After restricting RCB to 70, RR reached 71 runs by losing four wickets and 42 balls to spare.
One run out, two wickets by Mitchell Starc and one by Yuzvendra Chahal had given RR few jolts in their run chase but that were not enough to restrict Virat Kohli's side from reaching the small target.
Ajinkya Rahane (23), Shane Watson (24) made some crucial contributions with their willows.
Earlier, RCB were all out for 70.
Tambe took four wickets by giving away 20 runs which destroyed the RCB batting.
Virat Kohli remained the top scorer for his side with 21.
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