The 36-year-old, the eighth highest run scorer in Tests, had never reached three figures at the home of cricket and achieved one of the few milestones missing from his glittering career after lunch.
Sangakkara went to three figures by smashing part-time spinner Joe Root through cover before being embraced by close friend Mahela Jayawardene.
The pair put on 126 for the third wicket before Jayawardene fell for 55 but Sangakkara and captain Angelo Mathews saw off the new ball to steer the tourists past their initial target of 376.
Sangakkara fell nine runs later, giving debutant all-rounder Moeen Ali his first Test wicket.
Mathews was 79 not out at the close while Rangana Herath was unbeaten on nought.
Sri Lanka started the day on 140 for one in reply to England's mammoth 575 for nine declared after the 23-year-old Root had made 200 not out.
On a flat pitch that had few demons despite gloomy overhead conditions, the visitors lost opener Kaushal Silva (63) early on, having added a single run to his overnight score.
The diminutive batsman had already been troubled by one James Anderson bouncer and then failed to get his bat down while evading another, allowing Matt Prior to grab a sharp chance behind the stumps.
Sangakkara, who scored 319 and 105 in his last two Test innings against Bangladesh in February, brought up his 50 with a lovely push through the covers off Liam Plunkett.
Jayawardene, two places ahead of his partner on the all-time run scoring list, was less fluent but still managed to fight his way through to a 47th Test half-century.
He fell leg before wicket to Stuart Broad after England had taken the new ball while Lahiru Thirimanne went for two, clipping Anderson to debutant Sam Robson at square leg.
Anderson and Broad bowled with commendable endeavour in the middle session but as the ball began to age, Sangakkara and Mathews, who brought up his 50 off 64 balls, increased the scoring rate on a pitch that appears increasingly unlikely to produce a positive result.
Sangakkara was particularly strong off his hips and was removed three runs short of 150. He attempted to cut Moeen and feathered the ball through to Prior after spending more than a day in the middle.
Image: Kumar Sangakkara
Photograph: Gareth Copley/Getty Images
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