A 43-run opening stand between Graeme Smith (19) and Quiney (51) set the stage for a late charge and even though Yusuf Pathan (17) did not click today, Jadeja's 32-ball knock ensured the Royals had a decent total to defend even though they looked like getting 160 at some stage.
Rajasthan skipper Shane Warne, nursing a hamstring injury, caught everyone by surprise when he walked out for the toss which he won and opted to bat first.
The idea was clearly to shun extravagance early in their innings as the left-handed duo of Smith and Quiney seemed happy rotating the strike and helping themselves to the occasional boundaries.
Batting on 10, Smith got a life when JP Duminy spilled a tough chance at point off Dhawal Kulkarni, a lapse the youngster later atoned for.
Duminy's innocuous off-spinners troubled Smith, prompting Mumbai captain Sachin Tendulkar to press Harbhajan Singh into service and decision immediately paid off.Smith went for a heave that found Duminy in the deep to give him his moment of redemption. Yusuf Pathan (17) sizzled before he fizzled out, Lasith Malinga's well-judged catch off Chetanya Nanda ending his 14-ball stay.
Jadeja hit Harbhajan for two fours in the same over before hitting Duminy out of the park, a treatment he meted out to Bravo too.
His aggression rubbed on Quiney as well and the Victorian hit Bravo for a six and followed it up with a four to bring up his maiden IPL fifty before a Malinga yorker crashed through his defence.
Malinga bowled well at death again and it was largely to his stifling accuracy that Rajasthan eventually settled for a score below their own projection.
Malinga's compatriot Sanath Jayasuriya also bowled a brilliant last over, claiming two wickets, including Jadeja's, for just three runs.