New Zealand captain Mitchell Santner's destructive spell (4-21) had left India shaken on Saturday, with 68 runs needed from the last five overs.
But Varma (51) and Nitish Kumar Reddy (26) targeted the pacers in the final three overs to secure a four-wicket win.
"The Hyderabadi boys were in play today ... what everyone believed, I have done it, so I'm really proud of myself," said player-of-the-match Varma, who added 69 off 30 balls with Reddy in a brilliant unbroken seventh-wicket partnership.
The Christchurch pitch looked slower than in Thursday's high-scoring contest as India's pacers restricted New Zealand to 3-43 during the powerplay, but Glenn Phillips (70) and Bevon Jacobs (58) turned the innings around, adding 116 for the fourth wicket.
Phillips scored his second-straight fifty as he hit four boundaries and four sixes before Arshdeep Singh (2-33) got him caught near the fence.
Michael Bracewell then clocked 10 off two balls to help the hosts reach 5-180.
India's reply got off to a shaky start with Sanju Samson (14) playing out a maiden in the first over before edging Jacob Duffy's slower ball to point.
Vaibhav Sooryavanshi, who replaced the injured Ishan Kishan, scored four before falling to Nathan Smith.
Santner's left-arm spin added to the damage, sending back opener Abhishek Sharma (27) before bowling India captain Shreyas Iyer (7) with a flighted ball.
Two balls later, Axar Patel nearly cleared the long on boundary but Finn Allen jumped to catch it over his shoulder, tossing it to Daryl Mitchell before he landed over the rope.
Naman Dhir, who had helped close out India's chase on Thursday, fought back with a quick 31 off 17 balls, but he too fell to Santner, caught at extra cover to leave India reeling at 112-6 in the 15th over.
But Varma, in at number six, hit Duffy for two boundaries before punishing Smith with three fours and a six, getting India within 11 runs of victory with one over left.
Varma hit Duffy for four off the second ball of the final over to ease the pressure, before Reddy brought up the winning runs with a shot down mid on.
India can seal the series with a win at Wellington on Tuesday.
