Sri Lanka reached 134-4 in 17.5 overs to complete the win after restricting India to an unimpressive 130-4.
Former captain Sangakkara ensured there was no heartbreak this time after Sri Lanka had lost in four ICC world finals over the past seven years, including the World Cups in 2007 and 2011.
Sri Lanka was making its third World Twenty20 final appearance. It lost to Pakistan in the 2009 final and to the West Indies in 2012.
Sangakkara saw the team through in the company of fellow left-hander Thisara Perera, whose seam bowling was not used earlier in the day despite Perera replacing legspinner Seekkugge Prasanna.
Sangakkara and veteran teammate Mahela Jayawardene had announced that they would not take part in international T20s after this tournament. Sangakkara made it a memorable final outing for both with a fluent 35-ball knock in which he hit six fours and one six.
The Indian innings could never get the impetus it needed since defending a total less than 150 was always going to be difficult.
Kohli struck 77 off 58 balls with five fours and four sixes but some tight bowling from Sri Lanka restricted the free flow of runs expected from the Indian batting lineup.
The departure of a well-set Rohit Sharma for 29 and a slow 11 off 21 by Yuvraj Singh saw the Sri Lankan bowlers seize the moment.
Kohli, who survived two chances on 11 and 65, completed his fourth half-century of the tournament to finish as the highest scorer of this World Twenty20 with 319 runs and was also named player of the tournament.
Kohli was dropped early on by Lasith Malinga at mid-wicket off left-arm spinner Rangana Herath while Sachithra Senanayake failed to judge a catch on the mid-wicket fence off Nuwan Kulasekara to let the ball loop over his head and go for four.
India could score only 19 runs in the last four overs as Malinga and Nuwan Kulasekara bowled out beautifully, going for yorkers and full-length deliveries outside the off stump.
India was in line to hold all three ICC tournaments at the same time since it won the 2011 World Cup and the 2013 Champions Trophy.