So, England take a 1-0 series lead with two matches left to play.
Thanks for your company this evening. We'll be back on Saturday for the second ODI at Southampton, which starts at 11:00 BST. I'll leave you with highlights of Eoin Morgan's match-winning century.
In a bit!
'I've not been batting or bowling well recently'
And here is man of the match Moeen Ali, speaking to TMS.
"To get us to 339 in the end, I was very pleased with that. We finished well with the momentum.
"We bat all the way down so I gave myself a bit of a chance. I backed my game a bit more today - in the past sometimes I've not done that.
"I've not been batting or bowling well recently. Paul Farbrace gave me a few tips with my batting and that really helped me today.
"A lot of guys were very upset with what happened in Manchester and we dedicate this game to them."
Post update
South Africa's tail had a bit of fun, with Rabada and Parnell striking a few late boundaries, but Eoin Morgan reintroduced Chris Woakes to the attack to round off the innings and secure a 72-run victory.
Post update
However, Miller repeated Duminy's shot to exactly the same effect, before Chris Morris made a horrible tangle of a reverse sweep...
De Villiers kept hitting out but Moeen got the upper hand...
Post update
Du Plessis and Amla put on 112 runs for the second wicket but Amla was trapped lbw by Mark Wood, then eight balls later, Liam Plunkett snared Faf du Plessis with a nice bit of seam bowling...
AB De Villiers played nicely but his partners at the other end weren't playing so sensibly. JP Duminy fell to a reckless shot, but David Miller hit this towering six - which ended rather embarrassingly for one fella in the crowd..
Post update
Despite losing Quinton de Kock early, South Africa got off to a pretty decent start. Hashim Amla was in typically classy form, as was Faf du Plessis, and England's bowlers toiled for a little while...
And then Hashim Amla brought up his 32nd - 32nd! - one-day half century in a nicely poised innings...
Post update
Right, shall we have a little recap?
'A good day at the office'
England captain Eoin Morgan, who scored 107 off 93 balls, says: "It was a pretty good day at the office for us. You can't be sure what's a good score on here.
"I have felt good for a while. I didn't play as much in the IPL as I would have liked but I got a few runs in the Ireland game and to continue that here is a good feeling."
Asked how happy he is that he picked Moeen Ali, Morgan says: "You know what, if we didn't get many runs or wickets today, I probably would have still picked him. I think he's a fantastic player!
Man of the match - Moeen Ali
An unbeaten 77 with the bat and 2-50 with the ball. Moeen Ali is the man of the match.
Post update
Ebony Rainford-Brent
Ex-England batter on BBC Test Match Special
Wood coming on and taking the key wicket of Amla really changed the course of the game. After that, South Africa really struggled and lost wickets at regular intervals.
'We didn't bat well enough'
South African captain AB De Villiers on Sky Sports: "England played phenomenally well with the bat in hand. Morgs really dominated with the hundred and manipulated us really well.
"We didn't bat well enough, we definitely should have come a lot closer. We just didn't get partnerships going tonight.
"We lost Faf and Hash in close succession and that probably stopped us a little bit. After those two we never really got a partnership going that could have got us a lot closer.
On the Champions Trophy: "We all have a bit of pressure on us and we're all contenders to win that trophy. They played better tonight and hopefully we'll win the next one."
ReutersCopyright: Reuters
Post update
South Africa will rue letting those chances slips. A few messy dismissals in and around some very good batting. Amla and Du Plessis had an excellent partnership and De Villiers looked in good touch, but there were some careless shots.
Post update
Ebony Rainford-Brent
Ex-England batter on BBC Test Match Special
Eoin Morgan has the awareness of the game and the gaps. He'll have a gameplan within seconds, and he'll free his arms. He's in fine form leading in to that Champions Trophy.
Get Involved
#bbccricket
Rob Carrick: Cracking win for @ECB_cricket Solid win & room to improve with the bat, will take good side to beat us at the ICC Champs Trophy
Post update
Some good stuff from England there; a fabulous century from Morgan and some top class hitting from Moeen at the end. With the ball, England were a little stymied when Du Plessis and Amla were at the crease, but South Africa allowed them a way back into the game.
Post update
Ebony Rainford-Brent
Ex-England batter on BBC Test Match Special
In tough times sometimes you need something to give you a smile. It was tough start to the day with the minute's silence for the Manchester bomb victims but the boys started the game in the right spirit.
Post update
South Africa were 145-1 at one point. England's bowling was good but South Africa gave those middle-order wickets away.
Post update
Ebony Rainford-Brent
Ex-England batter on BBC Test Match Special
England have still got a lot of room for improvement but I'm very impressed with how they've approached today.
WICKET - England win by 72 runs
Rabada c Buttler b Woakes 19 (SA 267)
Game over! England win! Woakes finishes with four wickets as Rabada nicks him behind. A grateful Buttler holds on to it, and England take a 1-0 series lead!
WICKET
Parnell c Wood b Woakes 19 (SA 267-9)
Woakes does the job for Eoin Morgan - Parnell has a swipe at a short ball but can only send a leading edge high into the Headingley sky. Mark Wood darts forward and takes the catch before galloping off for a high-five with his teammates.
Live Reporting
Amy Lofthouse
All times stated are UK
Get involved
ReutersCopyright: Reuters 
Latest PostGoodbye!
So, England take a 1-0 series lead with two matches left to play.
Thanks for your company this evening. We'll be back on Saturday for the second ODI at Southampton, which starts at 11:00 BST. I'll leave you with highlights of Eoin Morgan's match-winning century.
In a bit!
'I've not been batting or bowling well recently'
And here is man of the match Moeen Ali, speaking to TMS.
"To get us to 339 in the end, I was very pleased with that. We finished well with the momentum.
"We bat all the way down so I gave myself a bit of a chance. I backed my game a bit more today - in the past sometimes I've not done that.
"I've not been batting or bowling well recently. Paul Farbrace gave me a few tips with my batting and that really helped me today.
"A lot of guys were very upset with what happened in Manchester and we dedicate this game to them."
Post update
South Africa's tail had a bit of fun, with Rabada and Parnell striking a few late boundaries, but Eoin Morgan reintroduced Chris Woakes to the attack to round off the innings and secure a 72-run victory.
Post update
However, Miller repeated Duminy's shot to exactly the same effect, before Chris Morris made a horrible tangle of a reverse sweep...
De Villiers kept hitting out but Moeen got the upper hand...
Post update
Du Plessis and Amla put on 112 runs for the second wicket but Amla was trapped lbw by Mark Wood, then eight balls later, Liam Plunkett snared Faf du Plessis with a nice bit of seam bowling...
AB De Villiers played nicely but his partners at the other end weren't playing so sensibly. JP Duminy fell to a reckless shot, but David Miller hit this towering six - which ended rather embarrassingly for one fella in the crowd..
Post update
Despite losing Quinton de Kock early, South Africa got off to a pretty decent start. Hashim Amla was in typically classy form, as was Faf du Plessis, and England's bowlers toiled for a little while...
And then Hashim Amla brought up his 32nd - 32nd! - one-day half century in a nicely poised innings...
Post update
Right, shall we have a little recap?
'A good day at the office'
England captain Eoin Morgan, who scored 107 off 93 balls, says: "It was a pretty good day at the office for us. You can't be sure what's a good score on here.
"I have felt good for a while. I didn't play as much in the IPL as I would have liked but I got a few runs in the Ireland game and to continue that here is a good feeling."
Asked how happy he is that he picked Moeen Ali, Morgan says: "You know what, if we didn't get many runs or wickets today, I probably would have still picked him. I think he's a fantastic player!
Man of the match - Moeen Ali
An unbeaten 77 with the bat and 2-50 with the ball. Moeen Ali is the man of the match.
Post update
Ebony Rainford-Brent
Ex-England batter on BBC Test Match Special
Wood coming on and taking the key wicket of Amla really changed the course of the game. After that, South Africa really struggled and lost wickets at regular intervals.
'We didn't bat well enough'
South African captain AB De Villiers on Sky Sports: "England played phenomenally well with the bat in hand. Morgs really dominated with the hundred and manipulated us really well.
"We didn't bat well enough, we definitely should have come a lot closer. We just didn't get partnerships going tonight.
"We lost Faf and Hash in close succession and that probably stopped us a little bit. After those two we never really got a partnership going that could have got us a lot closer.
On the Champions Trophy: "We all have a bit of pressure on us and we're all contenders to win that trophy. They played better tonight and hopefully we'll win the next one."
Post update
South Africa will rue letting those chances slips. A few messy dismissals in and around some very good batting. Amla and Du Plessis had an excellent partnership and De Villiers looked in good touch, but there were some careless shots.
Post update
Ebony Rainford-Brent
Ex-England batter on BBC Test Match Special
Eoin Morgan has the awareness of the game and the gaps. He'll have a gameplan within seconds, and he'll free his arms. He's in fine form leading in to that Champions Trophy.
Get Involved
#bbccricket
Rob Carrick: Cracking win for @ECB_cricket Solid win & room to improve with the bat, will take good side to beat us at the ICC Champs Trophy
Post update
Some good stuff from England there; a fabulous century from Morgan and some top class hitting from Moeen at the end. With the ball, England were a little stymied when Du Plessis and Amla were at the crease, but South Africa allowed them a way back into the game.
Post update
Ebony Rainford-Brent
Ex-England batter on BBC Test Match Special
In tough times sometimes you need something to give you a smile. It was tough start to the day with the minute's silence for the Manchester bomb victims but the boys started the game in the right spirit.
Post update
South Africa were 145-1 at one point. England's bowling was good but South Africa gave those middle-order wickets away.
Post update
Ebony Rainford-Brent
Ex-England batter on BBC Test Match Special
England have still got a lot of room for improvement but I'm very impressed with how they've approached today.
WICKET - England win by 72 runs
Rabada c Buttler b Woakes 19 (SA 267)
Game over! England win! Woakes finishes with four wickets as Rabada nicks him behind. A grateful Buttler holds on to it, and England take a 1-0 series lead!
WICKET
Parnell c Wood b Woakes 19 (SA 267-9)
Woakes does the job for Eoin Morgan - Parnell has a swipe at a short ball but can only send a leading edge high into the Headingley sky. Mark Wood darts forward and takes the catch before galloping off for a high-five with his teammates.