The Black Caps, set a Duckworth-Lewis target of 216 in 33 overs, were 126-6 in 25 overs when bad light ended play early in this central Sri Lankan town.
The tourists, who should have been 162 at that stage, were declared losers by 36 runs as per the D/L method to determine rain-affected matches.
Sri Lanka, sent in to bat, had made 211-8 from 33 overs after the start was delayed by 90 minutes and a further rain interruption reduced the game to 33 overs each.
Left-arm spinner Rangana Herath claimed three wickets and off-break bowler Sachithra Senanayake took two as the tourists slumped to 63-6 by the 19th over.
James Neesham and Nathan McCullum smashed 63 for the seventh wicket in just 36 deliveries to lead a fightback before the umpires called off play.
Neesham was unbeaten on 42 and McCullum was on 35 off 19 balls.
Earlier, Tillakaratne Dilshan hit 53 and put on 91 for the first wicket with Mahela Jayawardene, before Sri Lanka slipped to 146-6 against the steady Black Caps attack.
The ninth-wicket pair of Senanayake and Herath boosted the total by smashing 18 runs in the final over bowled by seamer Andrew Ellis.
New Zealand will round off their short two-week tour with two Twenty20 internationals in Pallekele on November 19 and 21.
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