Rayudu smashed 63 runs off 37 balls with the help of six fours and three sixes after opening batsman Tiwary set the foundation with a 43-ball 52, which was studded with nine boundaries.
For KKR, Trent Boult (2/30) took two wickets, while Ankit Rajpoot (1/14) and Kuldeep Yadav (1/25) took one wicket each after skipper Gautam Gambhir won the toss and elected to bowl in the match which was delayed due to rain and thunderstorm.
Earlier, Kolkata made one change, bringing in Trent Boult in place of Chris Woakes, who is out with an ankle niggle, while Mumbai decided to test their bench strength and made as many as six changes in their playing XI.
Ambati Rayudu, Saurabh Tiwary, Krunal Pandya, Vinay Kumar, Tim Southee and Mitchell Johnson were included in the MI team, while Lasith Malinga, Nitish Rana, Parthiv Patel, Mitchell McClenaghan, Jasrprit Bumrah and Harbhajan Singh were rested for the game.
Mumbai Indians: 173 runs in 20 overs (Ambati Rayudu 63, Sourabh Tiwary 52; Trent Boult 2/30).