Chasing a modest 157 for only their second win, Sri Lanka overcame the loss of two early wickets before Pathum Nissanka (77 not out) and Sadeera Samarawickrama (65 not out) got them across the finish line in 25.4 overs.
They earlier restricted England to a below-par 156 all out in 33.2 overs in the must-win encounter between the tournament's stragglers, with only Ben Stokes offering some resistance in an otherwise dismal batting display.
Stokes (43) took the attack to Sri Lanka on a dry wicket at the M Chinnaswamy Stadium after England's middle-order collapsed in the face of some incisive bowling, but he holed out to Dushan Hemantha in the deep off Lahiru Kumara (3-35) in the 31st over.
"I'm very happy with my performance, I've worked really hard for it," said man of the match Kumara, before heaping praise on returning veteran Angelo Mathews for his impact.
"Angelo has a lot of experience, he helped me a lot at mid-off. It was really great to have him back."
England enjoyed an early slice of luck after Jos Buttler won the toss and opted to bat as Jonny Bairstow (30) survived an lbw chance when Sri Lanka failed to review, but he did not build on his breezy start and gifted his wicket to Kasun Rajitha (2-36).
Sri Lanka had put the brakes on England's scoring following the powerplay as veteran Angelo Mathews (2-14) dismissed opener Dawid Malan (28) and then helped run out Joe Root before Kumara struck twice to leave the holders in tatters at 85-5.
Mathews, called in to play in his fourth World Cup as an injury replacement for Matheesha Pathirana, snared Moeen Ali in his second spell while Chris Woakes fell to Rajitha after a low diving catch by Samarawickrama.
"The plan was to hit the middle overs with discipline, we stuck to it and were rewarded with wickets," Kumara said.
Sri Lanka, who came into the contest with the same record as England - three losses in four matches - then ripped through the tail to ensure the lowest all-out total in Bengaluru and lay the platform for a comprehensive victory.
"It's been an incredibly tough and incredibly disappointing tournament," said Buttler.
"Disappointed in myself and all the boys that we haven't shown a good account of ourselves. There's no clear answer at the moment. Can't really fault the boys' efforts, but we're playing a long way short of our best."
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