Sri Lanka, chasing an improbable 660 for victory, were 231 for six at stumps on the fourth day, with Dilruwan Perera (22) and Suranga Lakmal (16) set to resume on Sunday needing to bat the entire day to save the match and series.
The first game in the two-match series ended in a damp draw in Wellington last week.
Sri Lanka had been set the target late on Friday after Tom Latham (176) and Henry Nicholls (162 not out) drove New Zealand's second innings to 585 for four declared.
The tourists lost two early wickets in their response but had resumed on Saturday on 24 for two with captain Dinesh Chandimal on 14 and Kusal Mendis on six.
Despite a solid start, their hopes of saving the test effectively ended in the session before tea with the dismissal of Mendis and an injury to Angelo Mathews.
Mendis, who helped save the first test with Mathews with an unbroken 274-run partnership, appeared well set to repeat the dose with Chandimal on Saturday as they thwarted New Zealand for the entire first session.
However, the 23-year-old Mendis, who joined India captain Virat Kohli as the only players to surpass 1,000 test runs in 2018, fell after lunch for 67 to a superb diving catch by Matt Henry at mid-off.
The way in which the New Zealanders mobbed the substitute fielder showed how much they valued the wicket, which ended a 117-run partnership between Mendis and Chandimal.
Mathews, who top scored with 33 not out in Sri Lanka's first innings of 104, looked like he would be even more difficult to remove than Mendis before he pulled up after completing a run and immediately clutched at his left hamstring.
He received treatment on the field but was clearly hampered and could do nothing but play from the crease and hop when Chandimal called him through for a run.
The former captain did not return after the tea break and is expected to have scans later on Saturday, although it is doubtful he will bat on Sunday.
(Reporting by Greg Stutchbury; Editing by Peter Rutherford)
