The tourists went to lunch at 38-2 at the University Oval and added only 65 runs in the two hours and 30 overs of the second session, maintaining an overall run rate of only 2.17.
Opener Dimuth Karunaratne kept up a careful vigil, reaching his seventh test half century in just under three hours and going on to be 59 at the interval.
Dinesh Chandimal was 31 not out as the pair added 74 runs in 35.1 overs since the fall of Sri Lanka's second wicket at 29-2.
The slow pace of Sri Lanka's scoring was due in part to deliberate watchfulness from Karunaratne and Chandimal, as well as the accuracy of the New Zealand bowling.
Tim Southee conceded only 16 runs from 10 overs at tea, Doug Bracewell allowed 22 runs from his 12 overs and left-arm spinner Mitchell Santner conceded just 12 runs from 10 overs.
The Sri Lanka run rate at times fell below two runs per over but Karunaratne and Chandimal picked up the pace a little in the second hour of the second session. Karunaratne had hit six boundaries at the break, playing especially strongly through the on side while Chandimal used his feet well against the seamers and against Santner who sought to tie him down.
The pitch at the University Oval was heavily-grassed and promised assistance for fast bowlers when Sri Lanka won the toss and chose to field. But while there was moderate pace and bounce, there was little seam movement and clear overhead conditions produced little swing.
The Sri Lanka batting lineup has a new look after the retirements of Kumar Sangakkara and Mahela Jayawardene, and there was uncertainty about how well it would fare in what, for most batsmen, were unfamiliar conditions.
Udara Jayasundera, who is making his test debut, was out for 1 before lunch while Kusal Mendis, in only his second test, went for 8. Both fell to catches by wicketkeeper B.J. Watling which carried him past 100 dismissals in tests.
Pressure fell on Karunaratne and Chandimal as two of the most experienced batsmen in the current lineup and they responded with a stoic partnership, which staunched the early flow of wickets.
New Zealand had earlier been dismissed for 431 after resuming Friday at 409-8. Bracewell carried his score from 32 overnight to 47 — his highest score in tests — before being the last man out.
Nuwan Pradeep wrapped up the New Zealand tail, taking 2-11 on the second day to finish with 4-112.
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