Asked to bat first, Daredevils made 152 for five while RCB, cruising at 129 for two at one stage, all but committed hara-kiri as they lost five wickets in a matter of 17 deliveries leaving them to get 12 off the 20th over bowled by Irfan Pathan.
After West Indian all-rounder Ravi Rampaul hit a six off the first ball and a couple of singles later, RCB required two off the last.
Rampaul and Vinay Kumar scrambled for a bye even as wicket keeper Kedar Jadhav fumbled with the collection as the match ended in a tie leading to the Super Over.
AB de Villiers struck two sixes off Umesh Yadav as RCB made 15. In reply, Daredevils lost David Warner to the first ball bowled by Rampaul.
However, Irfan Pathan slammed a six and a boundary off the second and fourth balls, followed by a single off the next. RCB clinched the game when Rampaul clean bowled Ben Rohrer off the final delivery.
Earlier, skipper Virat Kohli slammed his fourth half-century (65, 50b, 7x4, 1x6) in six matches, but RCB made a hash of the chase before coming through in the Super Over.
It was RCB's second Super Over game, having lost to Sunrisers Hyderabad earlier this month following a tie.
Kohli was at his dazzling best after RCB lost both their openers Chris Gayle (13) and streaky Lokesh Rahul (12) for 26 runs and in the company of AB de Villiers (39, 32b, 3x4, 1x6) put on 103 for the third wicket to script a victory.
Rahul, lucky to survive a confident leg before appeal in the very first over bowled by Ashish Nehra, struck a couple of boundaries before falling to the same bowler.
Gayle, after smashing Morne Morkel for two sixes, was dismissed by the crafty South African fast bowler, holing out in the deep attempting another big hit.
Then came the entertaining partnership between Kohli and de Villiers. Both were in imperious form as RCB purred along with neither batsman in any sort of bother until the latter was run out as Morkel's direct hit caught him short of the crease while attempting a sharp single in the 16th over.
In the very next over, Andrew McDonald departed without scoring, offering a tame return catch to spinner Shahbaz Nadeem as RCB slid to 129 for four.
Worse was to follow for RCB as Arun Karthik (5) was run out attempting a fourth run, and then seamer Umesh Yadav dismissed Syed Mohammad (0) and Kohli in the 19th over off successive deliveries to give Daredevils a sniff of victory.
However, the seventh wicket pair of Rampaul and Vinay Kumar kept its nerves to ensure a tie and the Super Over.
Earlier, David Warner (15) and Virender Sehwag (23) provided a rollicking start to the Daredevils innings. Both were at their aggressive best with a series of boundary hits and looked extremely dangerous.
However, both were dismissed within four balls as Warner was caught and bowled by Vinay Kumar who showed good reflexes to grab an uppish shot, while in the very next over, Sehwag flicked McDonald's first delivery straight to Kohli at mid-wicket.
The two dismissals put the Daredevils on the backfoot and they could never really regain the early momentum with skipper Mahela Jayawardene (28) struggling to break the shackles while Manpreet Juneja (17) and Ben Rohrer (14) failed to stay long enough to build meaningful partnerships with the Sri Lankan maestro.
Jayawardene himself was dismissed when he was caught well outside the crease backing up as Kedar Jadhav's straight drive struck the stumps off bowler Vinay Kumar's foot.
Eventually, Jadhav (29 not out off 16 balls) and Irfan Pathan (19 not out off eight) used the long handle to good effect and took Daredevils to 152 for five with the last four overs producing 47 runs.