Cook had no answer for those who doubt his limited-overs credentials, making just five in his comeback after 15 months out, but positive batting from Craig Kieswetter (61) and a devastating spell by James Anderson (four for 18) ensured his reign had a victorious start.
With each innings reduced to 32 overs after a lengthy rain break, England posted an imposing 229 for eight before bundling the tourists out for 121 in pursuit of a Duckworth/Lewis-revised target.
The hosts showed five changes from the side that was routed by nine wickets in Saturday's Twenty20 match at Bristol, with Ian Bell, Anderson, Jonathan Trott and Tim Bresnan as well as Cook returning to the side.
Sri Lanka's chase got off to a woeful start, with both openers gone inside 10 balls.
Fit-again Sri Lanka captain Tillakaratne Dilshan fared even worse than his opposite number, caught at fine leg off Anderson's third ball for one. And there was no dream farewell for Sanath Jayasuriya, two days shy of his 42nd birthday and playing a 445th and final ODI.
He took two singles before slashing Bresnan to point where Eoin Morgan held a smart catch. Jayasuriya was awarded the standing ovation his fine career deserved.
For Anderson, omitted entirely from the Twenty20 squad, there was even better to come. In his third over he removed Sri Lanka's best two players, Mahela Jayawardene and Kumar Sangakkara, in the space of four balls. Jayawardene was lbw for five before Anderson pouched a neat return catch to send back Sangakkara
That made it 15 for four and exposed Sri Lanka's batting line-up, though Angelo Mathews was off the mark with a powerful drive to the long-off boundary.
Debutant Jade Dernbach almost struck with his first delivery, Mathews thick-edging one a yard short of Stuart Broad at third man. He did not have to wait long, though, striking eight balls later when Anderson took a superb catch to see off Mathews for 16. At 40 for five the contest appeared over and at the halfway stage of the innings Sri Lanka had crawled to just 55.
Jeevan Mendis (nine) chipped out off Graeme Swann before Anderson made Nuwan Kulasekara (five) his fourth wicket.
Dernbach ended Thilina Kandamby's docile knock of 19 in 48 balls to make it 69 for eight.
Suraj Randiv (24) and Lasith Malinga (26) shared a boundary-strewn stand of 52 for the ninth wicket before Swann removed both batsmen to wrap things up.
England's day and indeed Cook's tenure, had earlier started in frustrating fashion.
Having talked up his ability to succeed in the format before the game, Cook took five off his first two balls before feathering the third to Sangakkara behind the stumps.
Trott settled quickly with three fours but showers settled in with the score on 35 for one after seven overs. When play eventually resumed at 5pm with 18 overs lost from the innings, Kieswetter came out of his shell, pounding Suranga Lakmal for six over mid-on.
Trott contributed a couple more boundaries before Lakmal trapped him lbw for 23. Kevin Pietersen survived a direct hit by Dilshan before both he and Kieswetter upped the rate again.
The first two overs of spin yielded 25 runs before Pietersen (26) drilled Mendis' loosener straight to midwicket.
Morgan took successive fours first off Mendis and then Lakmal as he scored fluently.
Kieswetter perished to Randiv for 61 off 56 balls, with Morgan next out, to Lakmal, after progressing to 45 in 35 deliveries.
Bell added 12 before Jayasuriya made him his final ODI victim, 21 years after Mohammad Azharuddin became the first.
There were 44 runs in the last four overs before Malinga accounted for both Bresnan and Broad.