The triumph was built on a sensational batting display after Sri Lanka slipped to 24 for two. Pavan Rathnayake and Avishka Fernando steadied the innings with a 166-run partnership for the third wicket, with Rathnayake eventually making 130 from 87 balls.
Fernando narrowly missed out on a century, falling for 97 after returning a catch to Jamie Overton.
Rathnayake's dismissal in the closing stages left Dasun Shanaka to take charge, and he blasted eight sixes in a 97-run innings. Shanaka also departed on 97, but Janith Liyanage held his nerve, finishing unbeaten on 61.
With eight runs needed from the final over, Liyanage struck a decisive six off Josh Tongue to seal victory with one delivery remaining.
England's enormous total was headlined by captain Harry Brook, who smashed 202 runs to register the fastest double century in ODI history. He reached the milestone in just 114 deliveries, surpassing Shubman Gill's previous record of 117 balls.
Brook accelerated dramatically after a cautious start, using inventive ramps, scoops and slices to dismantle the Sri Lankan attack.
Joe Root contributed 69 runs, becoming the first Englishman to reach 8,000 ODI runs, while Tom Banton scored another half-century and Will Jacks hammered an unbeaten 66 from 30 balls.
However, Dunith Wellalage's economical spell proved valuable for Sri Lanka, while England's bowlers could not consistently interrupt the scoring.
Tongue threatened to turn the contest around by taking two wickets in successive deliveries during the final over, but Liyanage's composure ultimately decided an extraordinary encounter.
Sri Lanka's successful pursuit ranks second in ODI history, behind the famous 438-run chase completed 20 years earlier.
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