Requiring to chase down a 190-run target within 15 overs, Mumbai Indians overhauled the target in 14.4 overs when Aditya Tare hoisted James Faulkner for a six.
In a heart-stopping finish, when scored were tied in 14.3 overs, Tare pulled off one of the most memorable strokes of his career with that six that helped his side sneak into the play-offs.
It was a humongous task to score 190 runs in 14.3 overs as the target was initially but Anderson, who holds the world record of scoring fastest hundred in One-dayers, showed his worth with one of the most destructive knocks in IPL.
His unbeaten 95 runs came off 44 balls with nine fours and half-a-dozen sixes.
With this win, Mumbai edged out Royals on Net-Run-Rate to book a Eliminator match against Chennai Super Kings on Wednesday at Braborn stadium here.
Anderson did what Yusuf Pathan did to Sunrisers Hyderabad yestreday.
Mumbai were 61 for 3 and looking down the barrel but left-handed Anderson and Ambati Rayudu shared a game-changing 81-run stand for the fifth wicket.
Rayudu blasted 30 runs off just 10 balls.
Tare sounded the death-knell for Royals with a first ball six off left-arm quick Faulkner, who conceded 54 runs without taking a wicket. Royals bowling let them down badly.
Earlier, Sanju Samson and Karun Nair hit came up with responsible knocks to help Rajasthan Royals post an imposing 189 for four after being put into bat.
Samson scored 74 off 47 balls while Nair took 27 balls for his 50-run knock as the two young batsmen shared a 100-run stand in 57 balls after early departure of Shane Watson.