The match, delayed by 90 minutes due to bad weather, had been earlier truncated to 43 overs-a-side before another wet spell reduced it further to 33 overs each.
Sri Lanka, who were 91-0 at one stage following Dilshan's opening partnership with Mahela Jayawardene, slipped to 110-4 and then 146-6 against the steady Black Caps attack.
Nathan McCullum removed Jayawardene (46) and Kumar Sangakkara in three balls, before left-arm seamer Mitchell McClenaghan dismissed Dilshan and skipper Angelo Mathews off successive balls.
Sri Lanka need a win to draw the series 1-1 after New Zealand won the second game in Hambantota on Tuesday. The first match was rained off.
New Zealand will round off their short two-week tour with two Twenty20 internationals in Pallekele on November 19 and 21.
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