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The Black Caps messed up a simple chase of Sri Lanka's 138 all out and were reduced to 122-8, before the last-wicket pair of Tim Southee and Mitchell McClenaghan saw them through in the 37th over.
Left-arm seamer McClenaghan set up the victory with 4-43 after Sri Lanka won the toss and elected to bat, but Malinga stole the thunder with 4-34 that almost secured his team an improbable victory.
The dramatic win helped New Zealand end a six-match losing streak against the Sri Lankans and claim full points in their first match of the eight-nation tournament.
New Zealand were cruising at 48-1 when Sri Lanka hit back with three wickets for one run in the space of eight deliveries.
Malinga triggered the collapse with a full toss that crashed into Kane Williamson's pads. Umpire Rod Tucker upheld an appeal for leg-before.
Ross Taylor was leg-before to spinner Rangana Herath for zero, before Martin Guptill edged seamer Shaminda Eranga high to Mahela Jayawardene in the slips.
When James Franklin fell leg-before to Tillakaratne Dilshan in the 15th over, New Zealand had lost half their side for 70 runs.
Brothers Brendon and Nathan McCullum put on 35 for the seventh wicket to raise New Zealand's hopes before Malinga removed both batsmen in three balls to make it 122-8 in the 32nd over.
The ninth wicket of Kyle Mills fell when five runs were needed, and even though Malinga conceded just one run in the next over, Dilshan gave away four runs to seal the Black Caps' win.
Sri Lanka, who had won 10 of their last 11 completed 50-over games against New Zealand, let themselves down with a sloppy batting display in the first session.
Veteran Kumar Sangakkara played a lone hand for the Islanders with 68, his 75th one-day half-century, but five of the other six top-order batsmen failed to reach double figures as the team folded in 37.5 overs.
Sri Lanka never recovered after losing Kusal Perera off the first ball of the match when New Zealand captain Brendon McCullum flew to his left at second slip to pick up a brilliant catch off Mills.
Dilshan hit 20 off 18 balls when he was bowled by McClenaghan to make Sri Lanka 27-2.
It soon became 33-3 as veteran Daniel Vettori, playing his first one-dayer since the World Cup semi-final against the same opponents in Colombo two years ago, struck with his fourth delivery.
A slower, flighted ball caught star batsman Jayawardene on the back foot and umpire Bruce Oxenford upheld Vettori's loud appeal for leg-before.
Sangakkara fell in the 37th over, caught at point by Williamson off spinner Nathan McCullum, who finished with two wickets.
New Zealand next play Australia in Birmingham on Wednesday, a day before Sri Lanka clash with hosts England at the Oval in London.
Scores:
Sri Lanka: 138 all out off 37.5 overs (K.Sangakkara 68, M.McClenaghan 4-43);
New Zealand: 139 for nine off 36.3 overs (L.Malinga 4-34).
