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.
England finished on 286 for eight in just their eighth tie at this level, with Sri Lanka having thrashed them by nine wickets in their previous ODI encounter in Wellington at the 2015 World Cup. Woakes and Plunkett put on an unbroken 50 in 28 balls.
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.
The five-match series continues with another day/night clash at Edgbaston on Friday.