The off-spinner grabbed four wickets in a disciplined spell and then made a crucial 25 as England achieved the revised target of 164 in a game reduced to 48 overs a side due to rain to gain a 2-1 lead in the five-match series.
England were tottering at 107-7 on a difficult pitch before Swann added 40 for the eighth wicket with Stuart Broad (20 not out) to help his side win with seven balls to spare.
Swann, playing only his fourth one-dayer since making his debut in 2000, was named man of the match.
"Credit goes to all the boys. It was a big challenge batting under lights after losing the toss," said England captain Paul Collingwood.
"Broad has already scored vital runs for us and Swann showed a cool head. But we haven't won the series yet," he added.
Paceman Farveez Maharoof grabbed three early wickets and veteran left-arm spinner Sanath Jayasuriya took two wickets to raise Sri Lanka's hopes of victory, but Swann held his nerve under pressure on a slow pitc