Heavy rain overnight and throughout the morning saturated the playing area at Saxton Oval. When play began, the contest had been reduced to 24 overs per side and Sri Lanka won the toss and chose to bowl.
The tourists started strongly, reducing New Zealand to 75-3.
Big-hitting Martin Guptill was out for 27, made from 14 balls, Tom Latham fell for 9 and stand-in captain Kane Williamson was caught and bowled by Dushmantha Chameera for 12.
Umpires Richard Illingworth and Billy Bowden decided the arena could not be brought up to a reasonable playing standard in the time remaining.
The last match of the series will be at Mount Maunganui on Tuesday.
New Zealand won the first match by seven wickets, surpassing Sri Lanka's 188 in 21 overs, and the second by 10 wickets when it took only 8.2 overs to overtake the tourists' 117.