The tourists, set an improbable target of 413 following an unbeaten century by Tillakaratne Dilshan, plodded to 30-1 in 13 overs by stumps on the fourth day at the Galle International Stadium.
Dilshan made 123 not out as Sri Lanka declared their second innings at 259-4 at the stroke of tea.
A sharp shower during the tea break interrupted play by 40 minutes but New Zealand lost the wicket of Daniel Flynn before stumps when he was caught in the slips off Nuwan Kulasekara.
Martin Guptill was unbeaten on 17 and Ross Taylor was on eight when bad light stopped play with 14 overs remaining in the day.
Rain has been forecast during the final day's play on Saturday in which Sri Lanka have three sessions to bowl out the tourists and take the lead in the two-Test series.
New Zealand may be without influential batsman Jesse Ryder and wicket-keeper Brendon McCullum in their second knock after the duo suffered a stomach bug and were confined to their hotel rooms on Friday.
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