Their eight-wicket win over the Kochi Tuskers Kerala at Jaipur came thanks to Shane Watson (49) and Rahul Dravid (44) and the victory lifted them to second in the points table.
Shane Warne, their captain, said in a post-match television interview: "Our intensity was very good from ball one, the boys showed a lot of character today, it was a satisfying win.
"Watson is such a super player, arguably the best player in the world at moment. We don't look too much at the table, we are still at the halfway stage. We have four to five days off now, looking forward to that."
Warne led the Royals' revival on home soil with figures of three for 16 to finish off Kochi's top half, and Siddharth Trivedi then helped wrap up the innings for just 109 with his three for 19.
Rajasthan were superb in the outfield, with five catches and two run-outs.
Kochi were always struggling with the run-rate - after openers VVS Laxman (eight) and Mahela Jayawardene (13) departed early - only reaching 32 in six overs before Warne bowled fellow Australian Brad Hodge for eight.
Things improved slightly as Parthiv Patel (32 off 34) led a patient 48-run combination with Ravindra Jadeja (22 off 23) but Warne struck twice in successive deliveries to dispatch the duo and leave Kochi on 90 for five.
No other batsman was able to break into double figures subsequently as Trivedi stepped up to quickly dismiss Raiphi Gomez and Thisara Perera. He later accounted for Kedar Jadhav after Amit Singh had taken care of Vinay Kumar.
Dravid and Watson, the Royals openers, were in imperious form with the bat as they steadily took the total to 70 inside 11 overs.
Dravid played an uncharacteristically quickfire knock that included four boundaries and a six before being run out to give Kochi a glimmer of hope.
But Watson kept Rajasthan in cruise control, striking four boundaries of his own - not to mention two sixes off Muttiah Muralitharan - in a 34-run stand with Johan Botha (14 not out).
Watson was bowled short of his 50 by Jadeja but Botha was on hand to hit the winning boundary and spark the home celebrations.
Mahela Jayawardene, the Kochi captain, said: "It was going to be a tough task with the dew, we needed another 20-30 runs.
"We didn't bat well and they bowled well up front. Rahul and Watto showed their experience, played really well.
"After the start we had, I thought 120-130 was a good score, but we couldn't execute, [there is] plenty to learn from this. It is a close table, we need to keep focus."
* Compiled by The National staff, with agencies