The victory increases second-placed Kolkata's chances of making it to the knockout stages of the competition as they move five points clear of fifth place.
The key to the victory was the quick-scoring opening stand of 113 between Gambhir (56) and Brendon McCullum (42) which came off just 12 overs.
Gambhir said in his post match television interview: "It was important for us to take the advantage in the first few overs because we knew that the pitch was going to slow down."
The Kolkata innings fell away after the openers were dismissed, and Pune's attack, led by Angelo Mathews, who took two for 17 in his four overs, helped restrict Kolkata to 150 for five.
Pune slipped to 55 for five in their response and although Mathews and Sourav Ganguly, the Pune captain, reignited their chances with a 73-run partnership, their hopes were extinguished once Ganguly departed for 36, caught at mid-wicket by Iqbal Mathews off the bowling of Rajat Bjhatia.
Mathews, who had earlier thumped three consecutive sixes off Yusuf Pathan, fell in the last over after making 35 as Pune finished on 143 for eight, with Marchant de Lange the pick of the bowlers with three for 34, while Surine Narine went for a measly one for 13 off four overs.
Gambhir acknowledged that the bowlers had rescued his side after the batsman had failed to kick on from their good start.
"We need to do better in the second half of the innings," he said. "Lot of credit to the bowlers for the way they defended the low score."
A disappointed Ganguly said: "We lost too many wickets at the start, Narine bowled superbly, if he wasn't there we would have probably won the game.
"We need to bat well at the top of the innings and performing in the middle of the innings.
"We need to start winning."
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