England, set 287 to win, collapsed to 30 for four and 82 for six. But a stand of 138 between man-of-the-match Chris Woakes, who made an ODI-best 95 not out, and Jos Buttler (93) revived their chances.
After England had seen their hopes of victory disappear in a final over where they needed 14 to win, Plunkett drove the last ball from Nuwan Pradeep for a six over long-off to the delight of a capacity crowd at Trent Bridge.
Woakes and Plunkett put on an unbroken 50 in 28 balls.
"It was nice to get that one out," Plunkett told Sky Sports.
"Woakesy put us in a great position and we bat all the way down. It's good for the team... If you have a wobble up top you can carry on."
England captain Eoin Morgan added: "Jos Buttler, Chris Woakes and Liam Plunkett have got us out of a hole at the end.
Sri Lanka captain Angelo Mathews top-scored for his side with 73 and Seekkuge Prasanna blasted a quickfire 59 after England captain Eoin Morgan won the toss. Warwickshire all-rounder Woakes took two for 56.
Mathews then took two early wickets as England collapsed before they bounced back under the floodlights.
"I thought we had the game in the bag for most of the overs," said Mathews.
"Unfortunately the yorker fell just off line and went for a six. For Pradeep, it happens. Unfortunately for bowlers, it happens.
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