An opening stand of 43 by Lendl Simmons and Parthiv Patel gave Mumbai the platform to set a competitive score, with Rohit Sharma contributing 27 and Kieron Pollard adding 24.
Put into bat at the Wankhede Stadium, Mumbai set an imposing total of 187/5, which proved too much for Rajasthan to chase down despite an excellent 46-ball 76 from Sanju Samson
But things could have been different for Rajasthan had wicketkeeper Samson taken a straightforward catch to dismiss Simmons off Tim Southee???s bowling in the second over.
They were made to pay with Simmons and Patel hitting seven boundaries between them before the latter was caught and bowled by Dhawal Kulkarni for 23 with the last ball of the fifth over.
Simmons added two fours and a six to move on to 38 before Ankit Sharma dismissed him lbw with the first ball of the 10th over. Unmukt Chand followed him back to the hut in the next over when he was caught by Stuart Binny off Southee???s bowling.
Sharma was the next man in and he became only the second man after Sachin Tendulkar to score more than 2000 runs for the franchise when he took his score to 27, hitting Kulkarni over deep midwicket for six.
But Sharma was out next ball, with namesake Ankit taking a spectacular one-handed catch at short fine leg to leave Mumbai 120/4 after 15 overs.
Pollard joined Rayudu at the crease as the pair took the score to 139/4 by the end of the 17th. Rayudu hit three fours and a six, and Pollard two sixes and a four before the West Indian edged Juan Theron???s slower ball to Samson with just two deliveries remaining.
The batsman had crossed though, and Rayudu added a couple with the following ball before bringing his fifty up in style, cutting over short third man to take Mumbai on to 187.
Rajasthan started their innings with purpose but were also fortunate not to lose a wicket in the first over, Jagadeesha Suchith dropping a sitter to dismiss Ajinkya Rahane off McGlenahan.
The Orange Cap holder would go on to make 16 off 12 balls before Vinay Kumar dismissed him with the first ball of his spell when Rahane holed out to Rayudu.
But skipper Shane Watson led the Rajasthan charge, hitting three fours and a six to keep them ahead of the run-rate, taking the score to 49/1 from five overs.
Rajasthan were ticking along nicely until Suchith atoned for his earlier dropped catch by clean bowling Watson with the first ball of the ninth over. The Australian departing on 28.
But Samson and Steve Smith put Rajasthan in a good position, sharing a 56-run stand before the latter top-edged a short ball from McGlenahan to Harbhajan Singh at third man with the last ball of the 15th over.
The writing looked on the wall for Rajasthan when McGlenahan claimed his second scalp. Samson???s eye-catching innings ending when he picked out Suchith at long off, leaving the Royals needing 27 to win from the last 14 balls.
And McGlenahan claimed his third dismissal with the next ball, Nair edging it to Patel behind the stumps, reducing Rajasthan to 161/5.
Lasith Malinga dismissed Deepak Hooda late on before Southee was run out th the penultimate ball as Rajasthan finished nine runs short of the target.