England may now need to win their final Group A match against New Zealand on Sunday to reach the semi-finals, but current weather forecasts are unsettled.
England posted an impressive 293 for seven after losing the toss, Alastair Cook (59), Jonathan Trott (76) and Joe Root (68) making half-centuries before Ravi Bopara flogged 28 in the last over of the innings to massage the total.
But Sangakkara's 15th one-day hundred ensured that never seemed quite enough, finishing with 134 not out in 135 balls and finding joyous late support from pinch-hitter Nuwan Kulasekara, who belied an average of just 16 to make an explosive unbeaten 58.
Things had started relatively well for the home side, who will have been glad to see Root among the runs after the distraction of his bar-room altercation with David Warner over the weekend.
But their bowlers had no answer to Sangakkara, who came to the wicket in the third over after the impatient Kusal Perera fell early to James Anderson.
Sangakkara soon set about forging his first partnership of the day, putting on 92 runs in 18 and a half overs alongside the enthusiastic Tillakaratne Dilshan.
Cook had gambled on Root at first change but he was calmly milked for 27 runs by two excellent players of spin.
Sangakkara was scoring with real freedom as Sri Lanka eased into three figures but Dilshan (44) holed out to Graeme Swann after the spinner had dried things up.
That brought Jayawardene to the middle and soon after the umpires called for one of the balls to be changed, much to England's chagrin.
After the switch Jayawardene took a liking to Stuart Broad, who was taken for 20 in two overs, including a graceful straight six and two wristy pulls for four.
The balance of power was changing rapidly, a fact that was driven home when Anderson returned to the attack only to see Sangakkara edge his opening delivery for four past a vacant first slip.
The pair looked immovable at 187 for two but the powerplay immediately changed the game, Dilshan lofting Anderson to Jonny Bairstow at deep mid-wicket.
That left Sangakkara, on 90 not out, to take the side home but he almost found gully off Anderson's very next ball.
A gentle single took him to his century but celebrations were muted with 84 still needed.
Kulasekara was sent up the order to play some shots and he complied with aplomb, passing 20 at a run-a-ball before swiping successive sixes off Swann.
He followed by pulling Broad for a third maximum over long-leg and punched the next two balls for four to record a remarkable half-century.
Kulasekara's happy hitting took the pressure off Sangakkara, whose 11th boundary signalled the hundred partnership in just 68 balls.
His 12th won the game, with 17 balls remaining.
England openers Cook and Ian Bell began the day promisingly, blunting Lasith Malinga's first spell in a risk-free stand of 48.
It should have been more, but Bell departed for 20 when he carelessly flicked Shaminda Eranga to mid-wicket.
Cook would have followed had Dilshan held a sharp chance at backward point but instead set about building alongside Trott.
The pair nudged the ball around off spin and seam alike, but rarely looked to find the ropes in a second-wicket stand worth 89 in 17 overs.
Cook was dropped by Dilshan for the second time on 56 but added just three more before he was lbw sweeping Rangana Herath.
The incoming Root scored off his first eight deliveries, while Trott brought up a measured fifty in 63 balls.
The batting powerplay brought 37 runs at an improved rate, with Root also benefiting from the first two drops.
The Yorkshireman cashed in to reach his fourth ODI fifty and soon unveiled a remarkable reverse paddle off Kulasekara.
Trott's stay eventually ended in the 42nd over, lbw to Herath, and Sri Lanka used the opening to hit back.
Root and Morgan fell to successive Malinga deliveries, the latter to a poor lbw, before Jos Buttler nicked Kulasekara for a duck.
Just as England's innings looked to have sagged, Bopara stepped up. He clubbed Eranga for three sixes and two fours in a devastating final over.
Unfortunately for England that was merely a taste of what was to come in the second innings.