Sangakkara extended his recent glut of runs with the bat – having scored 67 against Pakistan in the tournament's opening game and 128 in his previous one-day international against Bangladesh – to ensure Sri Lanka made it two Asia Cup wins out of two.
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India valiantly attempted to defend their score of 9-264, with Mohammed Shami (3-81) and Ravindra Jadeja (3-30) both making an impact and the latter on a hat-trick at one point.
However, Sangakkara's masterful knock – which came off 84 balls and included 13 boundaries – proved the difference and set up a tight finish.
Earlier, Shikhar Dhawan (94) was again left to rue his misfortune as he fell marginally short of a century for the second time this month, during an India innings that ended with two huge maximums from Shami (14 from seven balls).
Having been dismissed for 98 in the recent Test series with New Zealand, Dhawan came up short of reaching three figures once more.
It was Ajantha Mendis (4-60) who eventually made the crucial breakthrough to remove Dhawan and then Dinesh Karthik (4) in the space of three balls in the 40th over.
Having won the toss and elected to field, Sri Lanka were dealt an early setback when Angelo Mathews was forced to leave the field for treatment mid-over with what appeared to be a groin injury.
Mathews did eventually return, but not to a Sri Lanka attack that initially struggled as Dhawan and Virat Kohli (48) put on a second-wicket stand of 97 after coming together with India on 1-33.
After Dhawan departed with the score having moved 4-196, wickets tumbled frequently in the final 10 overs as Mendis and Sachithra Senanayake (3-41) made light work of the tail, although Shami's late fireworks in the penultimate over helped India creep past the 250 mark.
Under the floodlights, openers Kusal Perera (64) and Lahiru Thirimanne (38) got Sri Lanka's reply off to a solid start before Sangakkara took centre stage.
The veteran brought up his 18th international ODI century in the penultimate over from a mis-field that went for four.
There was drama in the very next ball as the game hung in the balance when Sangakkara played a Shami full toss to Ravichandran Ashwin, who stooped to take a low catch.
Sangakkara asked the umpires to check if the delivery was over waist height, but following much deliberation, and after checking if the catch was cleanly taken, the 36-year-old was sent on his way.
Inspired by their team-mate Sangakkara's incredible composure at the crease, Mendis (5 not out) and Perera (11 not out) kept their cool under immense pressure to see Sri Lanka home with four balls to spare and inflict India's first defeat of this year's Asia Cup.