Leg-spinner Sharma returned figures of 4/16 - the joint best effort by a spinner in an IPL knockout game - while medium pacer Bumrah finished with 3/7 in three overs, as KKR were all out in 18.5 overs.
After Bumrah dismissed Chris Lynn for four in the second over, the 29-year-old Sharma emerged as the wrecker-in-chief, sending back KKR skipper Gautam Gambhir (12) and Colin de Grandhomme (0) off consecutive balls, besides castling Narine in the fifth over.
Ishank Jaggi (28) and Surayakumar Yadav (31) put on 56 runs for the sixth wicket in 47 balls, during which they belted five fours and one six, and saved KKR from further embarrassment.
KKR began the innings rather poorly, losing Lynn early, when he tried to loft Bumrah in the second over of the match.
Mumbai struck for the second time in the fifth over, removing Narine for 10, thanks to a superb googly by Sharma and wicketkeeper Parthiv Patel did the rest at the M Chinnaswamy Stadium.
In the very next over, Bumrah got his second wicket of the match, when he trapped Robin Uthappa plumb in front of the wicket.
The situation got worse for the Knight Riders as Sharma snapped up Colin de Grandhomme.
A face-saving partnership ensued before Mitchell Johnson claimed the wickets of Piyush Chawla and Nathan Coulter-Nile, leaving Kolkata at 101 for eight.
Bumrah and Lasith Malinga got the last two batsmen cheaply.
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