Opting to bat first Mumbai Indians put up a mammoth 194 for seven in 20 overs after Dwayne Smith struck 50 and Dinesh Karthik 43.
Their bowlers put on a great show, especially local boy and medium pacer Dhawal Kulkarni (3/19), to restrict Royal Challengers to 136 for seven in 20 overs. Dwayne Smith and Harbhajan Singh also picked up two wickets each.
Smith was adjudged as Man of the Match for his all-round performance.
But the credit must go to the new ball bowlers Mitchel Johnson and Lasith Malinga, who bowled an excellent line and length to tie down Chris Gayle.
Kulkarni gave Mumbai the first breakthrough as he dismissed Tilakaratne Dilshan (13) in the sixth over. Dilshan's horrid run in the season continued as his thick outside edge flew to Johnson at third man.
Much was expected from Gayle, who was coming fresh from his record unbeaten 66-ball 175 in the last game against Pune Warriors. But it wasn't the Jamaican's day. Gayle managed 18 off 20 balls before falling in the seventh over.
Harbhajan Singh (2/21) gave Mumbai the prized wicket of Gayle, who top edged and Ambati Rayudu took a superb catch at long-on. Harbhajan broke into celebrations as Mumbai had eliminated their biggest threat.
With Bangalore struggling at 37/2, Kulkarni crippled the batting as he got skipper Virat Kohli (1) and AB de Villiers (2), both caught behind by wicketkeeper Dinesh Karthik, that left the visitors at 43/4.
The battle was already lost and the lower middle-order put up some brief resistance. Saurabh Tiwary (21), Arun Karthik (12) and Syed Mohammed (9) all departed in quick successions and Royal Challengers were soon struggling at precarious 85/7.
Ravi Rampaul (23) and Vinay Kumar (26) came up with an unfinished stand of 51 that just helped them to reduce the margin of defeat.