Runako Morton was on 70 and Dwayne Bravo on 22 when players left the field midway through the second session. While the rain had stopped when the players took an early tea, poor light prevented any further play before the rain began again.
The visitors had been 224 for four at lunch after Brian Lara unleashed a flurry of boundaries to race through to 83 before he chopped on a delivery from Nathan Astle.
Lara, who had scored just seven runs in the series prior to the match in Napier, resumed on Sunday on 28 and immediately attacked, particularly against Shane Bond, with several boundaries carved out through aerial shots behind square on the off side.
The world record holder was just beginning to look dangerous and set for another of his trademark big scores when he bottom edged a pull shot on to his front leg with the ball ricocheting back on to the stumps.
Morton was fortunate to be there after a comical situation in which he was initially adjudicated run out when captain Shivnarine Chanderpaul failed to respond to a quick single with both players standing at the non-striker's end.
While Chanderpaul had not attempted a run, he had gone down far enough for the umpires to decide the pair had crossed, and with television replays showing Morton had beaten Chanderpaul to ground his bat, the West Indies captain was given out for two.
Daren Ganga, who had resumed on 31 with Lara, was bowled by Bond for 38 to leave West Indies at 111 for two after combining with Lara in a 74-run partnership.
New Zealand have already sealed the series after winning the first Test in Auckland by 27 runs and the second in Wellington by 10 wickets.