Two-time champions Kolkata posted 157-7 as Venkatesh Iyer top-scored with 42 off 21 balls after being invited to bat first at their home Eden Gardens in a rain-shortened 16-overs-a-side match.
Bowlers combined to defend the total after Mumbai raced to 65-0 and Chakravarthy, a mystery spinner who was named player of the match, got the big wickets of Rohit Sharma, who made 19 after coming in as impact substitute, and skipper Hardik Pandya, who was caught out for two.
Five-time winners Mumbai, who are already out of the play-off race, finished on 139-8 as Tilak Varma hit a 17-ball 32 before he fell to fast bowler Harshit Rana, who took two wickets in the last over.
Pace bowler Andre Russell also took two wickets, including that of Suryakumar Yadav, as table-toppers Kolkata got their ninth win in 12 matches to become the first team to book a play-off berth.
"I manifested before the game that we'd get through," Kolkata skipper Shreyas said.
"We were playing fantastic cricket, boys have taken responsibility. There have been many game-changers. The boys have shown great attitude and it was on display today. We have a mindset to win from any situation."
Kolkata are almost certain of a top-two finish.
The top four teams will make the play-offs, but teams number one and two will have the advantage of getting an extra match to enter the final on May 26 in Chennai.
Mumbai suffered their ninth loss in 13 matches and Pandya said: "Just want to go and enjoy, play good cricket. That's been the motto from the start but we haven't played enough good cricket."
Kolkata lost openers Phil Salt, out for six, and Sunil Narine, bowled for a duck off Jasprit Bumrah, in the first two overs and Shreyas soon fell for seven.
But Venkatesh, a left-hand batsman, made Kolkata hit back with six fours and two sixes and he set the pace that was picked up by Nitish Rana and Russell in a key stand of 39.
Rana, a left-hand batter, hit 33 and Russell smashed 24 off 14 before Rinku Singh, who hit 20, and Ramandeep Singh, unbeaten 17, helped Kolkata finish on a high.
Bumrah and veteran leg-spinner Piyush Chawla took two wickets each.
