The Kolkata team lost five wickets in the course of the run chase.
Yusuf Pathan played a blistering knock of 60 runs off 29 balls, hitting six fours and three sixes.
Pathan was also involved in a 96-run partnership with Andre Russell for the fifth wicket.
Russell scored 39 runs off 24 deliveries and launched the RCB bowlers for four maximums and a boundary.
However, they had a poor start, losing Robin Uthappa for 1.
Chris Lynn (15) clubbed a big six but failed to make use of the chance the Bangalore fielders provided him with.
KKR skipper Gautam Gambhir played a steady hand of 37 runs off 29 balls, hitting four fours and a six.
Chahal was the pick of the Bangalore bowlers, picking up 2 wickets. One each was picked by Binny, Aravind and Watson.
RCB scored 185 runs for the loss of seven wickets.
Chris Gayle, (7) who was making a comeback after the birth of his daughter, failed to make a contribution.
Virat Kohli's good run continues as he scored yet another half-century and gained back the Orange Cap from his Sunrisers counterpart David Warner.
Kohli scored 52 runs off 44 balls and hit four fours. He was also involved in a 84-run stand along with K L Rahul for the second wicket.
Rahul, who scored 52 runs off 32 deliveries, completed a hattrick of half-centuries. His innings was laden with six fours and two sixes.
Though AB de Villiers (4) failed to contribute much, Shane Watson played a handy knock of 34 runs in 21 balls and hit five fours and a six.
The RCB innings gained momentum in the last couple of overs as Sachin Baby (16 off 8 balls) and Stuart Binny (16 off 4 balls) found the fence with ease.
For KKR, Morne Morkel and Piyush Chawla picked up a couple of wickets each. Andre Russell and Umesh Yadav claimed one wicket each.
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