Nathan Coulter-Nile (4/14) and Jimmy Neesham (3/12) starred with the ball as Royals scored only 90 for nine from 20 overs.
Ishan Kishan hit 50 off 25 as Mumbai reached the target in just 8.2 overs.
The five-time winners will take fourth spot if they beat Sunrisers on Friday and Royals beat Kolkata Knight Riders on Thursday.
Royals drop to seventh in the table and while a top-four spot is not mathematically impossible, their far inferior run-rate all-but-ends their hopes of qualification.
Delhi Capitals, Chennai Super Kings and Royal Challengers Bangalore are all guaranteed a place in the playoffs.
Neesham and Coulter-Nile rip through Royals
After being put into bat in Sharjah, Royals got off to a flier and reached 21-0 after two overs, before it all went downhill.
Rajasthan Royals: 90/9 in 20 overs (Lewis 24, Coulter-Nile 4/14, Neesham 3/12, Bumrah 2/14)
Mumbai Indians: 94/2 in 8.2 overs (Kishan 50 n.o, Rohit 22). (BBC)