Sangakkara hit an unbeaten 52 as Sri Lanka reached India's 130 for four with six wickets in hand.
Sri Lanka's success prevented India from holding all three world limited overs titles at the same time.
Sangakkara and fellow veteran Mahela Jayawardene announced before the tournament that they would retire from international Twenty20 cricket after Sri Lanka's game.
Together they made sure it was a glorious exit as Sri Lanka won their first major title since the 1996 World Cup.
Thisara Perera hit the winning runs when he lofted spinner Ravichandran Ashwin back over his head for four with 13 balls to spare.
India had been forced to defend a below-par total despite Virat Kohli continuing his white-hot form.
The right-hander crashed 77 from 58 balls before he was run out from the final ball.
Kohli had been given a life on 11 when Lasith Malinga grassed a tough chance over his head at mid-wicket and he appeared set to make Sri Lanka pay in full.
Kohli helped to belt 16 from the 16th over, delivered by Nuwan Kulasekera, and at that point it appeared as though India were primed to launch a late assault.
It never arrived as just 19 were taken from the final four overs despite Kohli's presence.
He was frustrated by Yuvraj's Singh's 11 from 21 balls, before holing out off Kulasekara, before even Mahendra Singh Dhoni failed to supply the late hitting.
It proved crucial as Sri Lanka were always on target despite a nervy chase.
Jayawardene initially paced matters with 24 from as many balls, but when he was well caught off Suresh Raina at the half-way point the match was evenly poised.
Lahiru Thirimanne then feathered an edge behind off Amit Mishra and with the tension starting to rise Sri Lanka sent in Perera to pinch-hit their way out of trouble.
They hardly needed to as Sangakkara, who had struggled all tournament, picked up the slack to fire his side to victory.
Perera added the finishing touch, after an important 23 from 17 balls, but it was his veteran team-mate who took the plaudits on his final bow.
- Sports & Recreation
- Sri Lanka
- Mahela Jayawardene
- Thisara Perera
- Virat Kohli