Rachin Ravindra led New Zealand's charge as they scored 194 runs in the final two sessions of day four in pursuit of their target of 275, but a fast-deteriorating Galle surface and relentless pressure from Sri Lanka's spinners meant they have only two wickets in hand to notch the remaining 68 runs.
At stumps, Ravindra was unbeaten on 91, but he was fast running out of company; Ajaz Patel just about managed to survive through to stumps. New Zealand will, no doubt, fancy their chances of completing this chase, but it will not be easy when the nature of the surface is taken into account.
The tourists were 13-1 at lunch after opener Devon Conway dragged an Asitha Fernando delivery on to the stumps to depart for four.
Tom Latham, on five, and Kane Williamson, on four, were batting at the break.
This was a day, therefore, that belonged to the spinners, starting with Ajaz's five wickets in the morning session - he finished with 6 for 90 - and ending with Prabath Jayasuriya and Ramesh Mendis grabbing three apiece. In all, 14 wickets fell in the day, 13 of them to spin.
Patel posted the best Test-bowling figures for a New Zealander on Sri Lankan soil since 1998, when Daniel Vettori claimed 6-64 in Colombo.
A rest day was held Saturday because Sri Lanka voted in presidential elections and the match resumed without spectators due to a post-poll curfew still in effect on Sunday morning.
The hosts were 237-4 on their return and added another 72 runs, with veteran Angelo Matthews making 50 before offering a simple catch to first slip off Patel's bowling.
Dimuth Karunaratne top-scored for Sri Lanka with 83 before he was bowled by Patel on Friday.
New Zealand have never won in Galle in four prior Test matches. The second and final match of the series will be held at the same venue from Thursday.
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