The series decider went right down to the last ball after Zimbabwe restricted India to 138 for six and got close to the target before ending at 135 for six. Earlier, Kedar Jadhav bailed India out of trouble with a timely 58 off 42 balls.
Zimbabwe, needing 21 runs off the final over, were in touching distance of a famous series win until Barinder Sran held his nerve for a dramatic finish to the game.
A four was needed off the final ball but Chigumbura ended up hitting a low full straight into the hands of Yuzvendra Chahal at cover, triggering wild celebrations by the away side.
The second-string Indian team had to work a lot harder in the T20 series in comparison to three ODIs where they blanked Zimbabwe easily.
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The home team made a confident start to the run chase, reaching 40 for one in the first six overs.
Sibanda came out with an aggressive intent and put pressure on the bowlers especially Dhawal Kulkarni, who was hammered for three consecutive boundaries in his second over.
Sran, who ended with figures of 4/10 on his T20 debut on Monday, was not at his best today even though he provided the first breakthrough and held his nerve in the final over.
The hosts looked threatening with Sibanda in the middle but his and Masakadza's fall in quick succession turned the match in favour of India.
Peter Moor got into the act after Sibanda's dismissal, punishing a wayward Chahal for three sixes. Chahal offered a couple of long hops and Moor was quick to deposit them into the stands over mid wicket.
The leggie finally removed Moor but the target was very much within reach as Zimbabwe needed 51 off the last 36 balls.
(REOPENS FGN 32)
Earlier, Jadhav struck his maiden Twenty20 International half-century as India overcame a poor start to post a competitive total.
It was a golden opportunity for the young batsmen to make an impact in a must-win match but some of them ended up squandering it.
Opener Lokesh Rahul was put under pressure in the very first over as pacer Tendai Chatara bowled a rare maiden in this format.
Rahul was tentative at the start but bounced back in Chatara's following over, smashing him for a straight six and a couple of fours. Eventually, he did not last long and neither did his opening partner Mandeep Singh.
Rahul too was out to a lose stroke as he tried to whack Neville Madziva over mid-wicket, only to play on to his stumps.
With Manish Pandey run out after a brilliant direct hit from Elton Chigumbura, India were struggling at 29 for three in six overs.
Jadhav and Ambati Rayudu (20) were left to do the repair job and they took the innings forward with a 49-run stand off as many balls.
Captain M S Dhoni, who failed to get his team over the line in opening T20, was bowled by Donald Tiripano after making 9 off 14 balls.
It was a rather unusual dismissal as Dhoni, clearing his front foot for a big one, got an inside edge on to his stumps and one of the flying bails sneaked into his helmet to hit him on the right eye.
Jadhav held his end to reach his maiden half century and his innings was laced with seven fours and a six.
By playing this game, he equalled Ricky Ponting's record of captaining in 324 international matches to be at the top of the list alongside the Australian.
In came Kohli to rapturous chants of his name and got to
business straightaway, cutting a widish Santner ball to the boundary and then pulled Wagner for another leg-side boundary.
However, the Indian captain lasted only 10 balls. Wagner banged one short, which Kohli miscued a pull and found out Sodhi, who was specifically kept for that shot.
Earlier, Rahul made a cracking start, hitting two crisp boundaries off paceman Boult in the very first over.
Wagner kept banging it short to test the Indian openers while Boult preferred fullish length.
Rahul was confident and comfortable at the crease and never looked like troubled by either of the two fast bowlers.
Kiwi skipper introduced Santner in the eighth over and continued with Wagner from one end. Rahul swept the left-arm spinner for a six but the Kiwi bowler exacted his revenge by having the young Indian batsmen caught behind in the next over.