The win, which came after they chased down 131 in 17.5 overs, ended a run of defeats in recent global finals.
Sri Lanka lost in the 2007 and 2011 World Cup finals and 2009 and 2012 World T20 finals.
“I was pleased I finally did something for the team,” Sangakkara, playing his final Twenty20 international, said.
He came into the game having scored just 19 runs in five previous innings and has retired from international T20s with 1,330 runs in 55 matches.
India batsman and player of the tournament Virat Kohli made a superb 77 off 58 balls as Sri Lanka restricted his side to 4-130.
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