Sri Lanka knocked off their 277-run target with 22 balls to spare in a remarkable turnaround after being outplayed in the first two matches.
In addition to Dilshan's heroics, Lahiru Thirimanne cracked an unbeaten 87 and Danushka Gunathilaka made 65.
Dinesh Chandimal was not out 27 at the end as the match followed the same formula as the first two where the team that lost the toss batted second and dominated.
New Zealand now lead the series 2-1 with two matches remaining.
But they were more at home on the accommodating Saxton Oval where Dilshan took 92 balls for his 91 and stood in partnerships of 98 with Gunathilaka and 111 with Thirimanne.
New Zealand strike bowlers Tim Southee and Adam Milne were belted out of the attack in five overs.
Doug Bracewell was brought on in the sixth over to arrest the onslaught only to have Gunathilaka smack his first ball for six to bring up Sri Lanka's 50.
Dilshan was run out when turned back by Thirimanne when attempting a difficult single.
In contrast to Sri Lanka's rollicking performance, the New Zealand innings showed signs of rust after so few batsmen had a chance to get in the middle in the first two games.
Williamson, deputising as captain in place of the injured
Brendon McCullum, was dropped on 30 and added a further 29 runs before meekly yielding his wicket when he slapped a full toss from Milinda Siriwardana to Dushmantha Chameera at mid-on.
In addition to the valuable catch to remove Williamson, the 23-year-old Chameera claimed the wickets of Martin Guptill (30) and Henry Nicholls (20) and ran out Mitchell Santner (38).
When New Zealand looked to be lining themselves up for a big score on a wicket that suggested an innings of 300-plus, Vandersay removed Tom Latham for 42 and Ross Taylor without scoring in three deliveries.
They were the first ODI wickets for the 25-year-old leg-spinner playing in his second international.
Bracewell's career best 30 off 34 and Southee's 18 off four balls in the final over gave the New Zealand innings a late boost.
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