New Zealand spinner Ajaz Patel took 6-90 to dismiss Sri Lanka for 309 on day four of the first Test on Sunday with the visitors chasing 275 to win in Galle.
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 from four prior Test appearances.
The second and final match of the series will be held at the same venue from Thursday.