Anderson, level with Darren Gough on 234 before the game, took the wicket of Mitchell Marsh to move past the former Yorkshire seamer's old record.
The Lancashire seamer's record-breaking wicket came in the first ball of his seventh over at Edgbaston.
He went on to take his new record to 237 with two more wickets.
Six others have taken 100 ODIs wickets or more for England, but only Anderson and Gough have broken the 200 barrier.
Sri Lanka spinner Muttiah Muralitharan holds the world record with 534 wickets from 350 matches, while Pakistan pace bowler Wasim Akram took 502 from 356.
Anderson is 25th in the all-time world list but has only played 170 internationals.
The 30-year-old became only the fourth England player to take 300 Test wickets when he dismissed New Zealand opener Peter Fulton during the first Test at Lord's in May.
If he is to become England's most successful Test bowler of all time, he will have to overhaul Sir Ian Botham's 383, with Fred Trueman's 307 and Bob Willis's 325 his next targets.
In February of this year, Anderson claimed his 529th scalp in all formats to pass Botham's 528 and become England's leading international wicket-taker.
"He's right up there with the best that England have had and he's right up there with the best in the world," said former England captain Michael Vaughan last month.

More than 1,000 drivers and engineers were set to stage the walkout on Bank Holiday Monday in a dispute over rosters.
It was supposed to be the second day of strike action after Unite members at Tower Transit took action on Friday.
It said it decided to suspend Monday's industrial action after "solid support" for Friday's protest.
A spokesperson said it wanted to give the company time to "reflect and respond".
Transport for London said Tower Transit operated less than 5% of the bus network.

The top seeds defeated Australia's Dylan Alcott and South Africa's Lucas Sithole 6-0 3-6 6-2.
"It's an amazing feeling and a lovely end to a tough couple of days," said 24-year-old Lapthorne.
Meanwhile, fellow Britons Jordanne Whiley and Gordon Reid both reached their respective doubles finals.
Whiley and Japanese partner Yui Kamiji stayed on track to retain their women's doubles title after the Grand Slam-winning pair defeated Germany's Katharina Kruger and Dutchwoman Sharon Walraven 6-4 6-3.
In Friday's final they will face Dutch second seeds Jiske Griffioen and Aniek van Koot who they beat in the Roland Garros, Wimbledon and US Open finals last season, although the Dutch pair won their most recent encounter in the Sydney International Open.
"I'm really happy to make the final, but it wasn't my best day," said Whiley. "I certainly need to play better in the final."
Reid will contest his second Australian Open men's doubles final after partnering Argentina's Gustavo Fernandez to win their semi-final against second seeds Joachim Gerard of Belgium and Maikel Scheffers of the Netherlands 6-1 7-5, having come from 2-0 down and 5-3 down in the second set.
"I'm delighted to be back in the final after a good solid performance from us - we're pleased how we battled through the second set," said Reid, who lost in last year's final.
Reid and Fernandez will play defending champions and top seeds Stephane Houdet of France and Shingo Kunieda of Japan in Friday's final.
Earlier on Thursday, Lapthorne saw his hopes of a place in the quad singles final fade when he was edged out 6-3, 7-6 (7-4) by Sithole in his second of three round-robin singles matches.
James Anderson became England's leading wicket-taker in one-day internationals with his 235th victim as they
[NEXT_CONCEPT]
A planned bus drivers' strike which threatened to disrupt Notting Hill Carnival has been called off.
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Britain's Andy Lapthorne won his fourth Australian Open quad doubles title after he and world number one David Wagner retained their Melbourne crown.