Jasprit Bumrah (3-14) led India's tidy bowling display as they restricted New Zealand to a modest 153-9 after the hosts elected to field at the Barsapara Cricket Stadium.
India, who will be defending their title at next month's T20 World Cup, completed the chase in just 10 overs with Abhishek leading the charge and skipper Suryakumar Yadav making 57 not out off 26 balls.
Abhishek's 14-ball fifty, which included five sixes, was the second-fastest by an Indian - behind only Yuvraj Singh's 12-ball half-century against England in the 2007 T20 World Cup. Suryakumar needed only 25 balls to bring up his second successive fifty of the series.
Earlier, Hardik Pandya took a stunning catch to remove Devon Conway and dismissed Rachin Ravindra with his second ball as New Zealand slumped to 13-2 in the first 10 deliveries of their innings.
Bumrah's first delivery then sent Tim Seifert's off-stump cartwheeling.
Glenn Phillips (48) and Mark Chapman (32) halted the slide with a stand of 52 before spinner Ravi Bishnoi removed both batters to put India back in charge.
Kyle Jamieson lost his leg stump to Bumrah, who, operating as the second-change bowler, also claimed the wicket of New Zealand captain Mitchell Santner (27).
Matt Henry bowled Sanju Samson for a duck with his first ball when New Zealand came out to defend their below-par total on a high-scoring ground, but Ishan Kishan (28) hit the bowler for back-to-back sixes and followed up with a four.
Abhishek struck the first ball he faced, from Jacob Duffy, for six to signal India's attacking intent despite the early loss of Samson.
Kishan fell immediately after India had passed the 50-mark in the fourth over but Abhishek and Suryakumar plundered 22 runs from a Duffy over as India reached 100 in the seventh over.
The fourth game is in Visakhapatnam on Wednesday.