The players were forced from the field for the third and final time with New Zealand, batting first, 145-3 after 28.5 overs.
More than three hours later, the umpires ruled there was no likelihood of the weather clearing enough for play to resume and the match was abandoned.
For New Zealand Martin Guptill was not out 66, while for Sri Lanka Angelo Mathews had three for 21.
New Zealand were rocked early when batting mainstay Brendon McCullum was removed by Mathews for 28, putting pressure on Guptill to repay the faith the selectors had in him after he failed in the first two games.
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McCullum again blazed away in trademark fashion at the head of the innings, racing to 28 off 22 deliveries before mistiming a pull shot off Mathews's first delivery and edging the ball to Kulasekara at fine leg.
Tom Latham, another top order New Zealand batsman in need of confidence-boosting runs, proved an able lieutenant for Guptill.
He put on 42 in a 99-run stand for the second wicket before being bowled by Mathews when trying to settle in after the second rain break.
Taylor left the field as the rain swept in for the final time.
The seven-match series is locked at 1-1 with the fourth game to be played in Nelson on Tuesday.