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Shane Watson dismissed Gambhir in the sixth over to give Royals a crucial breakthrough and reduce Kolkata to 41/1 in 5.3 overs.
Asked to field, discipline Knight Riders bowlers took wickets at regular intervals to restrict Rajasthan to 132/6.
Royals batsmen failed to put on any substantial partnerships during their innings after losing two early wickets and set a modest 133-run target for the hosts KKR.
Sachithra Senanayake was the pick of Kolkata bowlers, taking two Royals wickets for 26 runs in his four overs. Rajat Bhatia (1-21), Sunil Narine (1-20), Iqbal Abdulla (1-22) and Jacques Kallis (1-25) also picked up a wicket apiece.
Young Sanju Samson was the top-scorer for the visitors with 36-ball 40 as he hit two maximums and a four in his innings. Shane Watson (35) and Owais Shah (24) also contributed handsomely in their team's effort.
In a surprise move, skipper Rahul Dravid dropped himself to No.7 and came to bat with just three balls remaining in the innings. He struck a boundary on the very first ball he faced but it was too late as Knight Riders had the perfect recipe in their effort to stay afloat in the competition.
The embattled Knight Riders went into the match with a three-pronged spin attack, bringing in Senanayake and left-arm spinner Abdulla alongside their trump card Narine.
It's a ploy that had given them results in the past on the Eden surface and they, once again, relied heavily on their spinners with their batters misfiring.
Earlier, Rajasthan Royals skipper Rahul Dravid won the toss and decided to bat against Kolkata Knight Riders in the defending champions' last league fixture at the Eden.
Rajasthan have made three changes in their team from the last game citing the slow and low wicket at the Eden. Brad Hogg replaced Brad Hodge, Ankit Chavan came in for Ajit Chandila and Dishant Yagnik included in Sreesanth's place.
Hosts KKR have included Sachithra Senanayake and left out Brett Lee for this game, whereas, spinner Iqbal Abdulla replaced pacer L Balaji.
Kolkata Knight Riders: Gautam Gambhir (Capt.), Manvinder Bisla (wk), Yusuf Pathan, Jacques Kallis, Eoin Morgan, Debabrata Das, Rajat Bhatia, Sumit Narwal, Sunil Narine, Iqbal Abdulla, Sachithra Senanayake.
Rajasthan Royals: Rahul Dravid (Capt.), Ajinkya Rahane, Sanju Samson, Shane Watson, Stuart Binny, Owais Shah, James Faulkner, Siddharth Trivedi, Brad Hogg, Ankit Chavan, Dishant Yagnik (wk).