Sangakkara, 37, scored a run-a-ball 112 after being dropped by England captain Alastair Cook on 41, a mistake that could cost the visitors dearly.
England must win the game to keep their chances alive of pocketing the seven-match series which they currently trail 3-2.
Sangakkara, playing his last international match in his home town of Kandy, hit two sixes and 12 fours before he was caught by Moeen Ali off paceman Chris Jordan (2-68).
Sangakkara shared a stand of 153 runs with Tillakaratne Dilshan (68) for the second wicket, overcoming a sluggish start against a disciplined England attack.
After electing to bat, Sri Lanka lost Mahela Jayawardene early, the veteran batsman lobbing a simple catch to James Taylor for just five.
England bowlers wilted under pressure, enabling Sri Lanka to plunder 92 in the last 10 overs despite a promising effort with the ball in the first five overs.
The seventh and final game will be played in Colombo on Tuesday.