Smith hit a career-best 141 for his eighth one-day hundred but his team were still restricted to 301-9 while chasing a 324-run target.
The defeat also eliminated the hosts, who needed 313 to stay ahead of Sri Lanka on net run-rate.
“It’s hugely disappointing. I think it’s about giving your all, unfortunately we fell 22 runs short,” said Smith.
“Two of the games, we have given away over 300 runs. It’s difficult to chase that much regularly. You have to give credit to the way England batted. Anything around 300 we would have cruised in, but 324 was a step too far.”
England have won both of their games in the four-team Group B, while South Africa and Sri Lanka finished their league engagements with one win apiece in three matches.
New Zealand, who have one win in two matches, will take on England in the last group match on Tuesday.
England owed their highest one-day total against South Africa to man-of-the-match Owais Shah and Eoin Morgan, who gave a superb exhibition of power-hitting. Shah smashed 98 off 89 balls, while Morgan cracked a 34-ball 67.
“The way Paul Collingwood and Owais played the spinners was crucial, setting the platform for Morgan at the end. James Anderson (3-42) has been exceptional,” said England captain Andrew Strauss.
South Africa went for their shots early in their innings in a bid to achieve a stiff target, with Smith and Herschelle Gibbs putting on 42 in seven overs.
Smith, dropped on 82 by Shah at long-on off seamer Collingwood, sustained his team’s hopes with a gutsy knock. He reached his century when he cut Collingwood for his 10th four.
Smith, refused a runner after suffering from cramp later in his innings, put on 78 for the third wicket with AB de Villiers (36) and 64 for the next with Jean-Paul Duminy (24).
But the asking-rate kept climbing as the hosts needed 94 to win in the closing 10 overs. It was all over when Smith fell in the 47th over after hitting 16 fours in his 134-ball knock.
Shah was earlier in the limelight, putting England on course for a big total with a gem of an innings containing six sixes and five fours. He added a record 163 for the third wicket with Collingwood (82).
England’s previous best for the third wicket against South Africa in one-day internationals was 114 between Marcus Trescothick and Anthony McGrath in Manchester in 2003.
“I’d love to bat on this pitch everyday. It was a beautiful wicket to bat on,” said Shah.
The stage was set for the final onslaught after the Shah-Collingwood stand, with England plundering 92 in their last 10 overs.
Like Shah, stand-in wicket-keeper Morgan also flayed the South African attack as he hit five sixes and four boundaries in his second successive half-century.
England began aggressively after winning the toss as Strauss (25) and Joe Denly (21) added 48 for the opening wicket before falling in the space of three overs.
But South Africa’s joy was short-lived as Shah and in-form Collingwood kept gathering runs comfortably against both pace and spin to put their team in a strong position.
Shah was more aggressive of the two, playing handsome strokes all round the wicket. He was just two runs short of his second one-day century when he was caught behind off spinner Johan Botha.
Market rout deepens in Indian tycoon Adani’s shares; Adani Enterprises loses $26 billion in value since report; Falls after Adani pulled share sale, investors spooked; Analysts say signals confidence crisis in Indian market
There will be 2-, 3- and 4-bedroom units, and residents will have a dedicated courtyard with shaded play areas, a climbing wall, sensory equipment and much more
Company targets $300 million revenue from the new terminal
The value of transactions in the first month of 2023 recorded a surge of 178 per cent to over Dh28 billion, according to data provided by Property Finder