The home side, under the guidance of new team director Peter Moores, scored a mammoth 60 runs from the final four overs with Jordan hitting 38 from just 13 balls, while Buttler added 26 from 20 deliveries to push England on to 247-6.
Sri Lanka, already chasing a revised target, saw their innings disrupted by a second rain break, and were bowled out for 144 with three wickets each for James Tredwell (3-38) and man of the match Jordan (3-25).
England endured a miserable winter on tour, but Cook was delighted to see his team commence their five-match series on a positive note by thrashing star bowler Lasith Malinga all around the Oval.
“The lads played him [Malinga] really well,” said Cook.
“The extra fielder up makes it hard for any bowler at the end, but the way the lads struck the extra runs at the end made all the difference.”
England got off to a storming start with Gary Ballance scoring 64, while opener Ian Bell reached 50, but was dismissed with the final ball before a long rain delay.
Eoin Morgan and Ravi Bopara then fell cheaply, but a quickfire stand of 54 for the seventh wicket put England in command.
The writing was on the wall for Sri Lanka when James Anderson dismissed Lahiru Thirimanne with the fifth ball of the innings.
Wickets steadily tumbled with only Mahela Jayawardene (35) offering any real resistance.
England are set to play Sri Lanka on Sunday at Durham.