Captain Virat Kohli and Chris Gayle were sent to try and eclipse the 20 runs scored by the Sunrisers duo of Cameron White and Thisara Perera.
Despite the best efforts of the visitors, though, former Royal Challengers bowler Dale Steyn managed to send a final delivery that could not be hit for a six, ensuring a second successive victory for the IPL's newest franchise in what just was the third Super Over decider of the tournament's history.
Earlier, Kohli had won the toss and opted to bat but saw star hitter Gayle get out for just a single run in the second over as he cut a ball by Hanuma Vihari to wicketkeeper Parthiv Patel's gloves.
Tillakaratne Dilshan followed suit two overs later as his offstump was bowled by Ishant Sharma.
It was left for Kohli (46) and Moises Henrique to steady the innings, with Karun Nair caught lbw between the two in the seventh owner.
The duo were a bit more successful with debutant Henrique lasting longer in the innings as Sharma (3/27) and Steyn cleaned up the lower order batsmen - the Royal Challengers lost eight wickets as they set a target of 131 for the home side.
Having successfully defended a similar low target against Pune Warriors on Friday, Kumar Sangakkara's side knew that it would not necessarily be an easy run-chase.
And they started off poorly with Patel leaving as early as the second over of the innings, White misjudging a length ball to give Muttiah Muralitharan an easy catch and Sangakkara gifting Arun Karthik the second of his catches behind the wicket off a Jaydev Unadkat (2/24) delivery.
However, Hanuma Vihari was the stabling influence for the home side and notched an unbeaten 44-run innings that saw the Sunrisers, who were down to their final three wickets, tie the match in the final ball of the game.
Vinay Kumar (1/27), who was quite successful in the final over against Mumbai Indians on Thursday, was tasked with the responsibility of bowling the Super Over, having just moments earlier conceded only six runs in the last six balls of the normal game.
Unfortunately, the 29-year-old's luck ran out as White returned to smash two sixes and a boundary to help score an impressive 20 off the over.
And when Gayle and Kohli padded up to face Steyn, they could only muster 15 runs in reply, ensuring a first defeat for the Royal Challengers in the 2013 season.
The Bangalore side can, however, secure some revenge as they face the same opposition again at the M. Chinnaswamy Stadium in their next IPL encounter on Tuesday afternoon.
In Sunday's other game, the Kings XI Punjab got their IPL campaign off to a positive start, beating the Pune Warriors by eight wickets and with nearly eight overs to spare on Sunday.
Adam Gilchrist's side, unfancied ahead of the game, needed only 12.3 overs to score the 100 runs required after restricting a dire Warriors batting order to 99 for nine in their 20 overs in Pune.
The Kings' bowling attack put forth a superb collective effort, with Azhar Mahmood showing his experience with two wickets at less than five to the over, while Praveen Kumar bagged Manish Pandey and Ross Taylor.
None of the Pune batsmen could get going on what was a good batting track, with Abhishek Nayar top-scoring with 25 at the end, while the big names like Taylor, Marlon Samuels and Angelo Mathews failed.
The Punjab side reached the target without much fuss, and Gilchrist, at the age of 41, showed his enduring class with a quick 15. Mandeep Singh and Manan Vohra, on debut, did the bulk of the scoring though.
Mandeep was bowled by Rahul Sharma for 31, but Vohra did well in his first game and was unbeaten on 43 at the end, with David Hussey next to him.