The third match in the series at Jade Stadium was shortened to 46 overs and New Zealand's victory target was reduced to 110 from 113 under the complicated rules used for rain-interrupted matches.
The home side appeared set to saunter to victory after Sri Lanka were dismissed cheaply for 112 off 35.2 overs. But by the 15th over, New Zealand had tumbled to 70-6 and the result was in the balance.
Peter Fulton came in at number four and steadied the innings with an unbeaten 43, finally clinching the match with a massive pull shot for six. The tall 27-year-old hit a series of elegant drives and pulls as he compiled five fours and two sixes in his 57 ball innings.
Allrounder James Franklin, who led New Zealand to victory with a composed 45 in the previous match against Sri Lanka, was again on hand to score an important 15 not out on the way to the total, reduced using the Duckworth-Lewis method.
Captain Daniel Vettori said he felt New Zealand always had Sri Lanka under pressure, despite the collapse early in the New Zealand innings.
"Maybe we were a little aggressive in going after the runs," Vettori said.
"But Peter Fulton and James Franklin again showed some pretty cool heads."
Sri Lankan skipper Mahela Jayawardene said his side came close to winning, despite the woeful batting performance.
"We knew we didn't deserve to win this game, purely because of the way we batted in the morning," he added.
New Zealand's victory gives the home side a 2-1 lead with two matches left to play in the five game series.
The damp conditions favoured the bowlers and for Sri Lanka, master spinner Muttiah Muralitharan was always a threat, taking 1-18 from his nine overs.
Ruchira Perera, with 2-37 from five overs, and Dilhara Fernando's 2-34 from 6.3 overs took valuable wickets, but also gave away too many runs given the low target set for New Zealand.
Sri Lanka's batsmen never seemed at ease and the tourists were unable to gain momentum amid the rapid fall of wickets.
Poor shot choices were behind the loss of many of Sri Lanka's wickets as play was periodically interrupted by showers.
New Zealand's Michael Mason was a hero with the bat in the second One-dayer on Sunday when he hit the winning four runs off the last ball to level the series. On Tuesday, he shone with the ball, finishing with the best figures of 4-24 off nine overs in a tight spell.
Left arm spinner Vettori proved typically tightfisted with runs, ending with 2-7 off his 5.2 overs.
Wicketkeeper Brendon McCallum played brilliantly behind the stumps with three excellent catches and he helped James Marshall complete a deft run out.
Sri Lanka was staggering at 61-5 by the 16th over and they could manage only another 51 runs for the remaining five wickets.
New Zealand's sharp fielding performance maintained the pressure on Sri Lanka with Marshall and Vettori instrumental in two excellent run outs.