The tourists went to lunch on the final day at 110-5, with only faint hopes remaining of reaching the record 390-run target they were set to square the two-match series.
With captain Angelo Mathews out on the last ball before the break, Lahiru Thirimanne was left not out 11.
For New Zealand Doug Bracewell has two for 23 and Trent Boult two for 39.
Sri Lanka resumed the day at 45-1 and quickly lost nightwatchman Dhammika Prasad for six when he edged Boult to Jimmy Neesham at second slip.
But he fell cheaply to Boult, just as he had twice in the first Test in Christchurch which New Zealand won by eight wickets.
This time Sangakkara reached five with two scoring shots before getting the faintest of edges, barely discernable by the hotspot technology, and was caught behind.
Wicketkeeper BJ Watling did not appear interested in an appeal at first but New Zealand persisted and after multiple replays the third umpire ruled there was wafer-thin evidence in favour of a dismissal.
However, it counted against New Zealand when the review showed the delivery was a no ball.
Silva progressed to 50 before he was eventually out to Bracewell.
In the last over before lunch Mathews was brilliantly caught one-handed in the gully by a diving Kane Williamson.
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