Australia were 125-4 at the 25-over halfway stage of their innings, needing 201 more runs to reach a target of 326.
Warner, like Steve Smith playing his first match against South Africa since they received 12-month bans for a ball-tampering scandal in Cape Town last year, was 68 not out and Alex Carey five not out.
South Africa -- who cannot qualify for the semi-finals -- made 325-6, with skipper Faf du Plessis scoring 100 and Rassie van der Dussen a career-best 95.
Victory would see Australia, already through to the semi-finals, top the table and ensure a return to Old Trafford for a semi-final clash with fourth-placed New Zealand on Tuesday.
But defeat could leave Australia with a seemingly tougher last-four showdown against resurgent hosts England at Edgbaston on Thursday.
They should have lost Warner in the first over of their reply but Kagiso Rabada threw to the wrong end.
Australia captain Aaron Finch, however, made just three before he chipped Imran Tahir to Aiden Markram at short cover, sparking a typically exuberant celebration from the 40-year-old leg-spinner.
Usman Khawaja retired hurt on six with a left hamstring injury.
Smith, should have been run out on two but it was not a costly error with the star batsman lbw for seven to paceman Dwaine Pretorius.
But left-hander Warner went past 550 runs for the tournament, during his sixth fifty of this World Cup.
Marcus Stoinis (22) was superbly run out when Quinton de Kock back-flicked Rabada's throw onto the stumps and the keeper then held an excellent leaping catch to dismiss all-rounder Glenn Maxwell (12) off the fast bowler.
Earlier, Mitchell Starc equalled Australia legend Glenn McGrath's World Cup tournament record of 26 wickets, set at the 2007 event in the Caribbean, with a return of 2-59.
Plessis, who won the toss, and van der Dussen put on 151 for the third wicket in front of a sun-drenched crowd.
Aiden Markram (33), opening after Hashim Amla was ruled out with a knee injury playing football in a warm-up on Friday, and de Kock (52) set the tone in a first-wicket stand of 79 before they both fell to off-spinner Nathan Lyon.
Du Plessis, who drove fast bowler Pat Cummins for a superb six, completed the first hundred by a South African at this World Cup in 93 balls, with seven fours and two sixes.
But he was out to the next ball he faced, slicing Jason Behrendorff to third man.
Van der Dussen fell short of his maiden ODI hundred when, trying to hit the last ball of the innings for six, he holed out.
He faced 97 balls including four fours and four sixes, with the 30-year-old now boasting an impressive ODI average of 73.77 from 14 innings.