Reigning champions Australia looked set for a huge total at 185-2 but, one ball after completing his 100, skipper Aaron Finch holed out.
Australia continued to lose wickets but, with Alex Carey at 38 not out, they finished on 285-7 off their 50 overs to leave England with a stiff but not impossible chase.
However, an Australia attack led by left-arm quick Mitchell Starc defended a total of 288 to beat the West Indies by 15 runs earlier in the tournament, with Finch side's having won all four games in which they have bowled second at this World Cup.
Both Warner, the World Cup's leading run-scorer, and Steve Smith were booed as they entered and exited the field following their recent return from year-long bans for ball-tampering.
Only the top four at the end of the 10-team round-robin stage will qualify for the semi-finals. Australia are currently 2nd in the table. England are 4th but any more slip-ups for Morgan's men in their three remaining pool games against Australia, India and New Zealand could imperil their chances of reaching the knockout stages.
Top-ranked England, however, have won 10 of their past 11 ODIs against Australia.
England innings
England suffered a top-order collapse as they slumped to 26-3 in chase of the target set by the Aussies.
Jason Behrendorff struck with just the second ball of England's reply, an inswinger bowling struggling opener James Vince for a duck.
Fellow left-arm quick Mitchell Starc then took two wickets for 5 runs in 9 balls.
Test skipper Joe Root was plumb LBW to an inswinger for eight before England captain Eoin Morgan (four) fell into a hooking trap when a top-edge was held safely by Pat Cummins at fine leg.
Victory over their arch-rivals would see Australia, who have lost 10 of their past 11 one-day internationals against England, into the semi-finals.
And while a third defeat in the group stage would not spell the end of England's hopes of reaching the last four from the 10-team round-robin stage, it would set alarm bells ringing.
England's final three group opponents are Australia, India and New Zealand ─ teams they have not beaten at a World Cup since 1992.
Australia innings
Finch and David Warner (53) shared a fine first-wicket stand of 123, with Finch earning a reprieve on 15 when James Vince could not hold a tough chance at backward point.
The Australia skipper then looked to up the tempo with a legside 6 off Moeen Ali but the stand was broken in the 23rd over when off-spinner Moeen induced Warner to lob a gentle catch to Joe Root, running round from backward point.
Finch went to his century thanks to a misfield by Chris Woakes but next ball he miscued a hook off Jofra Archer and Woakes made no mistake with the catch at fine leg.
He was out for exactly 100, having faced 116 balls, with 11 4s and two 6s and his exit saw Archer become the tournament's outright leading wicket-taker with 16 scalps.
New batsman Glenn Maxwell hit a 6 and 4 off successive Archer deliveries only to fall for 12 when caught behind trying to 'ramp' Mark Wood.
Smith and Marcus Stoinis were then involved in a mix-up that saw Stoinis run out before Smith was caught in the deep off Woakes for 38.
"To get out straight after a hundred is never ideal," said Finch. "I think there is enough in the wicket. It is still a pretty good score. It wasn't the easiest wicket to start on. The ball was moving for the seamer. We were conscious to get through that and were able to. We could have got a few more runs but we have got to defend it regardless."
Toss, pre-match chatter
Hosts England won the toss and decided to bowl first against Australia.
"It's not a no-brainer, but we haven't seen much sun down south so it might do a bit this morning," said England Captain Eoin Morgan after choosing to bowl first.
Speaking of beating Australia 5-0 last year, Morgan said: "It builds confidence."
"For a long time we never won anything against Australia, but we have had some success recently and I think we have a good side.
"Jofra is good, he had stiffness in his side and we had a precautionary fitness test. We are confident," he concluded.
"I'm not too disappointed batting," says Aaron Finch. "I wasn't too sure about the wicket, but I would have probably bowled first as well. We've played consistent cricket throughout without being outstanding. If you go too hard you leave yourself open at the back-end against some world-class death bowlers."
"We aren't really bothered about how the crowd react. Every game is crucial and we can't take any lightly," says Finch, while referring to how the crowd might react.
England, bidding to win the World Cup for the first time, are unchanged from the side that suffered a surprise 20-run loss to Sri Lanka.
Australia made two changes from the team that Bangladesh by 48 runs in Nottingham, with off-spinner Nathan Lyon replacing leg-spinner Adam Zampa and Jason Behrendorff coming in for fellow paceman Nathan Coulter-Nile.
Team lineups
England go with the same team, Australia make two changes.
England
Jonny Bairstow James Vince Joe Root Eoin Morgan (capt) Ben Stokes Jos Buttler (wk) Moeen Ali Chris Woakes Adil Rashid Jofra Archer Mark Wood
Australia
Aaron Finch (capt) David Warner Usman Khawaja Steven Smith Glenn Maxwell Marcus Stoinis Alex Carey (wk) Jason Behrendorff Pat Cummins Mitchell Starc Nathan Lyon
