Playing in their first home game of the tournament in Mohali, Punjab lost the toss and were put into bat by the visitors.
However, both men were gone before the end of the 11th over and George Bailey (39) was the only other Kings XI player to score well as they finished on 156-8, with the tournament's leading scorer Glenn Maxwell notching just two before he was caught and bowled by Shreyas Gopal.
Lendl Simmons set the tone for the Indians right from the off, finishing on exactly 100 not out - the highest score of this year's IPL that included 14 fours and two sixes in 61 balls.
He provided the bulk of the runs, with Rohit Sharma's 18 the next highest score for the Indians as they reached their victory target with one over remaining.
Mumbai climb up two places to fifth in the table and now need to win all their remaining games to have a chance of reaching the playoffs.