Both sides scored 174 in their 20 overs after Sri Lanka were given out in the final ball of their innings to set up a one-over shootout for victory.
Tim Southee bowled to Mahela Jayawardene to get the super over underway, with Thisara Perera the other Sri Lankan batsman in play.
Sri Lanka profited from two wide balls by Southee to set a score of 13 for their over.
Lasith Malinga bowled to Brendon McCullum and Martin Guptill as they were set a target of 14 to win the match.
Malinga produced a disciplined over and dismissed McCullum thanks to a brilliant catch by Tilakaratne Dilshan to hand Sri Lanka a hard-fought victory.