New Zealand cashed in on a strong start provided by openers Rob Nicol (58) and Martin Guptill (38) to post 174 for seven wickets after skipper Ross Taylor had decided to bat first on a belter of a track at the Pallekele International Cricket Stadium.
Sri Lanka's response was even better as Dilshan and Sri Lanka captain Mahela Jayawardene (44) added 80 runs in about seven overs in the group one match.
Requiring eight runs off the last over sent down by Tim Southee, Sri Lanka managed seven to finish on 174 for six and forcing the Super Over where a team nominate three batsmen to face six deliveries from a rival bowler.
Sri Lanka scored 13 runs off Southee's over and then unleashed Lasith Malinga who conceded eight runs while removing Guptill to clinch the contest that brought alive the tournament.
Earlier, New Zealand were buoyed by an opening stand of 57 off 45 balls between Nicol and Guptill.
Nicol tore apart the Sri Lankan attack to score 58 off 40 balls and was severe on spinner Ajantha Mendis, hitting the spinner for 24 runs off his final over which included three sixes.
Sri Lanka's chase was off to a flying start as Jayawardene made a fluent 44 off 26 balls with three sixes and three fours and Dilshan struck five fours and three sixes in his 53-ball 76.
Even after Jayawardene fell, Sri Lanka appeared cruising towards victory but New Zealand effected two run outs to remove Dilshan and Kumar Sangakkara (21) to claw their way back into the contest.
Needing one run off the last ball, Lahiru Thirimanne (five) was dismissed by a freak run-out leaving the scores level.
Taylor failed to collect the return from the outfield and appeared to have dropped the ball but replays showed that he had knocked it onto the stumps and Thirimanne was found short of his crease.