5 overs at a packed Shere Bangla stadium.
Thisara Perera hit Ravichandran Ashwin for a winning boundary to seal the win, finishing with 21 not out.
But it was Sangakkara who anchored the chase with six boundaries and a six in his unbeaten knock.
The victory gave Sri Lanka their first world title in 18 years since winning the World Cup (50 overs) crown in 1996 and gave the World Twenty20 a fifth champion in as many editions.
India won the inaugural edition in 2007 followed by Pakistan (2009), England (2010) and the West Indies (2012).
It also broke Sri Lanka's jinx of losing the final of major world level events as they had lost two World Cup (50 overs) finals (2007 and 2011) besides being runners-up in as many World Twenty20 events in 2009 and 2012.
The victory also gave a memorable swansong to veteran Mahela Jayawardene and Sangakka