Chasing a target of 162, Royals won the game with five balls to spare as the former Australia captain anchored the innings with an unbeaten 59 off 48 balls. Riyan Parag (43 off 29 balls) also showed good temperament as he matched his skipper stroke for stroke during their 70 runs stand in 9.4 overs that paved the way for a comfortable victory.
Smith hit five boundaries and a six while Assam lad Riyan also had an identical number of hits to the fence.
This is Rajasthan's third win in nine games as they are now placed seventh in the league table above wooden spooners Royal Challengers Bangalore even though a play-off qualification looks a long shot as of now.
Earlier, Quinton de Kock continued his impressive run with another solid half-century as Mumbai Indians put up a decent 161 for 5 batting first.
De Kock smashed 65 off 47 balls with six fours and two sixes, adding 97 runs off 68 balls for the second wicket with Suryakumar Yadav (34 off 33 balls) that helped MI total gain a semblance of respectability.
Reinstated as skipper in place of Rahane, Smith introduced leg-spinner Shreyas Gopal (2/21 in 4 overs) in the third over. He deceived rival skipper Rohit Sharma (5), who stepped out only to offer a return catch after failing to read a googly.
Gopal could have got rid of De Kock off the last ball of the over but Jofra Archer dropped a difficult chance at long on. Making most out of this reprieve De Kock helped his team reach the 50-run mark in 6.5 overs.
He reached his own half-century in 34 balls and his partnership with Suryakumar was broken when latter was caught in deep off Stuart Binny in the 14th over.
A few balls later De Kock too was holed out in long-on as Hardik Pandya and Pollard joined forces with the scoreboard reading 112 for 3 in 15 overs.