After Kane Williamson and Henry Nicholls posted double centuries as the hosts made 580-4 declared on day two, it was the turn of New Zealand's bowlers to take centre stage at Wellington's Basin Reserve.
Spinner Michael Bracewell and seamer Matt Henry took three wickets apiece as the Sri Lankans were dismissed in their first innings before the tea break.
Despite a battling 89 from skipper Dimuth Karunaratne, Sri Lanka lost six wickets with the addition of only 55 runs after lunch.
With the threat of rain hanging over the fifth day of the match, Blacks Caps skipper Tim Southee wasted no time in sending Karunaratne and his opening partner Oshada Fernando straight back in to face the home attack again.
Fernando faced 36 balls before departing for five to give paceman Doug Bracewell, cousin of Michael, another wicket on his return to test cricket after a gap of nearly seven years.
Karunaratne played another captain's innings, however, recording a fourth half century in his last five innings and putting together a second-wicket partnership of 71 with Kusal Mendis.
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His New Zealand counterpart Southee separated them when Karunaratne, on 51, pulled a shot to deep midwicket where Devon Conway took a fine catch into the sun.
Angelo Mathews came out to join Mendis and the pair survived until stumps. Mendis will resume on Monday on 50 not out, while Mathews was unbeaten on one.
In one positive historical note for Sri Lanka, Mendis and Mathews both scored centuries as they batted together for two rain-disrupted days to save a draw the last time the two teams played a test in Wellington five years ago.
Michael Bracewell said New Zealand, who beat England last month after following on at Basin Reserve, would not be getting ahead of themselves.
"We've got a big job, the two guys at the crease at the moment are obviously very experienced and have scored a lot of test runs," he told reporters.
"You have that goal in the end, but really trying to concentrate on what you're doing in the moment."
