Stand-in captain Kyle Mills, who won the toss and elected to field, gave the tourists a flying start when he trapped opener Dimuth Karunaratne leg-before with the first delivery of the innings.
Dilshan and Sangakkara, however, settled in to pulverise the Black Caps during a second-wicket partnership of 137.
Sangakkara made 79 off 81 balls with nine boundaries and two sixes, while Dilshan scored 81.
Left-arm spinner Anton Devcich dealt a double blow when he removed Sangakkara and former captain Mahela Jayawardene in the space of four deliveries to reduce Sri Lanka to 139-3.
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Mills, who was named captain for the tour after regular skipper Brendon McCullum was rested, was New Zealand's most successful bowler with three for 49.
Mitchell McClenaghan and Devcich claimed two wickets each.
The Black Caps will play three one-day internationals and two Twenty20 games on their brief tour.