Bangalore set a winning target of 172, but the Royals stepped up to the challenge admirably with wicket-keeper batsman Sanju Samson smashing home an excellent 63, with strong support coming from Brad Hodge’s 32-run cameo.
Hodge, who stepped in after Samson was finally dismissed, and Shane Watson (41) saw the home side over the finish line with one ball to spare in a tense finish.
Samson became the youngest man to hit a half-ton in the IPL at the age of just 18 before he was eventually dismissed by Ravi Rampaul.
Bangalore’s Ranganath Vinay Kumar had helped set up a challenging chase for the hosts with a prolific final over putting the visitors at 171 for six after their 20 overs.
Watson chipped in with a crucial three wickets for the cost of just 22 runs off his four overs as the Australian all-rounder starred with the ball.
A dejected captain, Virat Kohli, said: "Another loss unfortunately. We were 10-15 runs short. Rajasthan batted really well. I got out at the wrong time, one batsman who could have added another 20-30 runs could have changed things.
"We got really excited looking at the wicket, didn't bowl at a good length. We bowled really short initially and gave many boundaries."
Later in the day, Mumbai Indians tallied 174 for three to beat Kings XI Punjab, who had set a target of 170 thanks mostly to David Miller's 56 and David Hussey's 34.
Rohit Sharma's unbeaten 79 was the highlight of the Indian's winning innings, all on the skipper's birthday.