Former captain Kumar Sangakkara was the only batsmen to offer any sustained resistance, scoring 68 from 87 deliveries with eight boundaries before he was the eighth man out.
Left-arm pace bowler Mitchell McClenaghan, who impressed in the 2-1 one-day series victory over England before the start of the tournament, took four for 43 off 8.5 overs.
New Zealand made a perfect start after losing the toss.
Captain Brendon McCullum leaped high to his left at second slip to take a two-handed catch when the first ball of the match from Kyle Mills caught the outside edge of Kusal Perera's bat.
McClenaghan clean bowled Tillakaratne Dilshan for 20 then Daniel Vettori captured a wicket in his first over in 50 overs cricket since the 2011 World Cup.
The 34-year-old left-arm spinner, who has been troubled by injuries, got his third delivery to Mahela Jayawardene to grip, turn and bounce over the off-stump. The next ball was slower and straighter, deceiving the batsman who was struck on the back pad and dismissed lbw for four.
McCullum continued to set attacking fields, denying the Sri Lankans the opportunity to take quick singles and rotating his bowlers constantly.
McClenaghan bowled Angelo Mathews for nine, knocking the leg stump over after the Sri Lanka captain had moved across his crease.
Lahiru Thirimanne received a life when wicketkeeper Luke Ronchi failed to gather a thick edge off Kane Williamson's off-spin but was then run out for nine after a horrible mixup with Sangakkara.
Thisara Perera survived a caught-and-bowled chance to Tim Southee and an lbw decision against off-spinner Nathan McCullum which was overturned when the television review showed the left-hander had got a thick inside edge on to his pad.
He did not linger much longer, lofting McClenaghan to Vettori at mid-off, while Sangakkara sliced Nathan McCullum to Williamson at short third man.
(Writing by John Mehaffey in London; Editing by Josh Reich)