Angelo Mathews and Prasanna Jayawardene were keeping New Zealand's attack at bay at lunch on the last day of the second and final Test Thursday but Sri Lanka still face a major fight to save the match.
All-rounder Mathews was 27 not out and Jayawardene 23 not out as the home side, chasing a 363-run target, reached 107-5 in their second innings at the break.
New Zealand, seeking their first Test victory in Sri Lanka in 14 years, have a minimum of 64 overs at the P. Sara Oval in Colombo to get the last five wickets and level the series.
The tourists consolidated their position on Wednesday when they took four quick wickets, including those of batting stars Mahela Jayawardene, Tillakaratne Dilshan and Kumar Sangakkara.
But they struggled in Thursday's morning session to finish off the hosts, who added 60 to their overnight total of 47-4.
Mathews and wicket-keeper Prasanna Jayawardene were watchful during their 44-run stand after their team lost the big wicket of Thilan Samaraweera in the opening hour -- run out after a mix-up with Mathews.
Samaraweera, who top-scored in the first innings with 76, had added just six to his overnight one.
Mathews and Prasanna Jayawardene then denied New Zealand for more than 23 overs. Mathews has so far faced 106 balls and his partner 69.
Sri Lanka lead the series 1-0 following their 10-wicket victory in the opening Test at Galle.