Gambhir's stay was short lived as he was bowled by Shivam Mavi leaving Delhi's run chase in trouble.
Nitish Rana scored a fluent 59 off 35 balls. In response, DD kept losing wickets at regular intervals and folded for 129.
But Kolkata came back hard and Delhi lost its last seven wickets for 43 runs in 5.5 overs.
Kolkata bowled out Delhi for 129 in 14.2 overs to register their second win of the season.
As Karthik rightly pointed out at the post-match press conference on Saturday, KKR's strength lies in their three-pronged spin attack led by mystery spinner Sunil Narine and also has celebrated chinaman Kuldeep Yadav and veteran leggie Piyush Chawla. The Australian tries to hit third six on trott to Kuldeep Yadav, but this time he holds it straight to Uthappa.
Trent Boult, who bowled a first-over maiden to Lynn, returned and clean bowled Russell with a yorker.
While young Rana never kept a wrong foot in his entire innings of 59 runs till his dismissal, Russell's innings sky-rocketed to 41 from 11 balls.
Delhi's chase stuttered at 24-3. A sharp glove-work from skipper Dinesh Karthik saw in-form Jason Roy stumped off Chawla in the first over.More news: Glasgow man arrested for attempting to use brother's identity
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Boult started off really well as he picked up a wicket and did not give a run in his first 10 deliveries.
Boult then had Narine caught behind in his second over, while when he snaffled Karthik (19) at square leg off Chris Morris in the 14th over, the hosts were 117-4, Robin Uthappa (35 off 19) and Chris Lynn (31) having blasted a second-wicket stand of 55. From there on, it was all downhill for Delhi as they lost wickets in a hurry. However, he departed soon after and the responsibility of carrying the innings forward was left to Nitish Rana.
WICKET! Trent Boult with the breakthrough, Narine edges one back to wicketkeeper Pant to deflects it to Maxwell on the first slip. And Kolkata Knight Riders won by 71 runs.
The captaincy mantle was handed over to Dinesh Karthik, who has so far looked out of sorts in the opening two games, where KKR finished second best to Chennai Super Kings (CSK) and Sunrisers Hyderabad (SRH), respectively.
The visitors won the toss at Eden Gardens and put Kolkata Knight Riders to bat.
Jason Roy, who was brought into the match against Mumbai in place of Colin Munro, will be looking to carry on the form after playing match winning knock in the last game.
He played a gem of an innings, selecting his shots with aplomb, as he hit five fours and four sixes to set up KKR's onslaught and allow Russell, who was dropped on 7 by Roy, to go for the big shots from the word go. Piyush Chawla, Andre Russell, Shivam Mavi and Tom Curran also took a wicket each. At one stage we thought they will not get to 170-175 but then getting to 200 was a good effort by them.