Needing 66 runs off 24 balls, KKR were up against it before Russell came in and smashed an unbeaten 48 to complete the job.
Russell's brutal hitting came after Virat Kohli (84 off 49) and AB de Villiers (63 off 32) recreated their "magic" with a delightful 108-run stand to take RCB to 205 for three.
The result means KKR have now won three out of their four games while RCB are have lost five in a row and are heading towards another sorry season.
KKR were kept alive in the chase through opener Chris Lynn (43 off 31), Robbie Uthappa (33 off 25) and Nitish Rana (37 off 23) before Russell singlehandedly got the job done in the death overs.
A pedestrian bowling effort from RCB pacers also helped KKR's cause. RCB spinners Pawan Negi (2/21) and Yuzvendra Chahal (1/24) bowled well and were able to generate sharp turn, putting brakes on the KKR scoring rate.
Earlier, Kohli and de Villiers hammered the bowlers at will during their entertaining stand. At one stage, it seemed RCB would get close to 250 but KKR did well to pull things back a bit on a batting beauty.
During his sublime knock, Kohli also overtook Suresh Raina to become the leading run-getter in IPL history besides completing 8000 runs in T20 cricket.
The RCB skipper hit some beautiful cover drives early on and his innings comprised nine fours and two sixes. De Villiers too was in his element, smashing five fours and four maximums.
Kohli was finally out in the 18th over off Kuldeep Yadav as he hit the ball straight into the bowler's hands. He was particularly severe on Prasidh Krishna, Lockie Fergusson and Nitesh Rana.
De Villiers played second fiddle to Kohli before cutting loose to produce some massive hits.
In the last over, Marcus Stoinis belted 18 runs off Prasidh with the help of two boundaries and a six, pushing the total past 200 mark.
Needing a good start desperately, Parthiv Patel and Kohli gave a good start for first time in the tournament, stitching 64 runs for the first wicket in 7.5 overs.
Patel was LBW to Nitesh Rana after scoring 25 off 24 balls with three boundaries to his credit.