Dilshan pressed on and seemed set to guide Sri Lanka to the win when he was run out, sent back and beaten by fine work by Ross Taylor at short fine leg, for 91 off 92 balls. He was nine short of what would have been a 23rd ODI ton.
Lahiru Thirimanne, after making a single in each of the first two games, hit the winning runs to finish on 87 off 103 balls, while Dinesh Chandimal, dropped by substitute fielder Ish Sodhi on 19 at long off, ended on 27.
New Zealand's early bowling was too loose and the fielders were unable to exert any pressure on the batsmen, with the runs flowing too easily.
In the first innings, Sri Lanka had put in a far more spirited performance in the field, and some canny bowling to keep New Zealand to the lowest total in the six ODIs played at this ground.
Sri Lanka's fielding display was more alert, although two catches were dropped, the field placings more aggressive as the tourists' sought to put a grim few days behind them.
The pitch was slower than at Christchurch's Hagley Oval and after a bright start, Sri Lanka's bowlers were able to get a decent hold on the run rate.
Martin Guptill's fine 2015 ended with a 30 off 28 balls, to finish the year with 1489 ODI runs, top of the chart.
Tom Latham and captain for the day, Kane Williamson - standing in for the injured Brendon McCullum -- put on 60 for the second wicket.
However the loss of Latham and Taylor in three deliveries from legspinner Jeffrey Vandersay put a brake on.
Williamson completed his 14th half century this year before slapping a full toss to deep mid on.
In the space of 20.1 overs, New Zealand lost five for 89, Mitchell Santner's 38 the pick of the middle order contributions.
At the 40-over mark, New Zealand had 206 for six, then lost Santner run out, due to smart work from bowler Dushmantha Chameera on his follow through.
New Zealand were scratching for late runs and managed just 34 in overs 44 to 49, however Tim Southee sent the last three balls of the innings for 6, 6 and 4 off Nuwan Pradeep to inject late impetus with 21 off the over.
Doug Bracewell's 30 off 34 balls was his best ODI effort while Adam Milne finished on 17, and Southee 18 off four balls.
Chameera was impressive in taking two for 38 from his 10 overs, to go with his run out, Vandersay, in his second ODI, took two for 55 as did Pradeep.
The fourth ODI is also in Nelson on Saturday.
SCOREBOARD
WAGONWHEEL
MANHATTAN