England's win left the five-match series all square at 2-2 ahead of Saturday's Old Trafford finale.
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England, chasing a revised target of 171 after rain reduced their innings by two overs, finished on 171 without loss with more than 24 overs to spare.
Craig Kieswetter was 72 not out.
Left-hander Cook's innings followed the 26-year-old's career-best 119 in a six-wicket defeat at Lord's on Sunday.
Man-of-the-match Cook, not even in England's squad at the World Cup this year on account of his supposedly slow one-day scoring rate, faced just 75 balls with 16 fours.
Kieswetter, who struck three sixes and eight fours in 68 balls, helped Cook set a new England record partnership against Sri Lanka, surpassing the fourth-wicket stand of 154 shared by Graeme Hick and Neil Fairbrother at Adelaide in 1998.
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England's victory -- only their fourth 10-wicket one-day international win -- was all but assured after they dismissed Sri Lanka for just 174.
James Anderson took three wickets f