Fellow Australian Riley Meredith and Jasprit Bumrah picked up one wicket each to leave the opposition reeling at 29 for five in the sixth over.
New Zealander Devon Conway and Moeen Ali were out for a duck while opener Ruturaj Gaikwad scored seven.
Chennai skipper MS Dhoni tried to put some resistance with his unbeaten 36 but failed to pile up runs as he kept losing his teammates at regular intervals.
Kumar Kartikeya removed West Indian all-rounder Dwayne Bravo for 12, followed by Simarjeet Singh's wicket in the same over.
Kumar returned with figures of two for 22. Meredith bagged two wickets while Ramandeep Singh and Bumrah took one each.
Mumbai also stuttered in the powerplay and were down by four wickets for 33.
Opener Ishan Kishan scored six while Sharma hit four 4s to reach 18 before he was caught behind by Dhoni off Singh.
Sams was caught leg before wicket by Mukesh Choudhary followed by Tristan Stubbs dismissal for naught in the same over.
But Tilak Varma and Hrithik Shokeen kept the scoreboard ticking before Hrithik fell to to Ali at 18. Tilak scored an unbeaten 34 while Tim David made a quick 16 off seven balls as Mumbai chased down the target with 31 balls to spare
Mukesh bagged three for 23 in his four while Simarjeet and Moeen took one each.
"The first few games didn't go to plan. Had a little bit of time to reflect on those performances. I just figured out that I was focusing on the batter too much and not on my strengths," said Sams, who won the player of the match award.
Chennai have lost eight matches in the tournament and are out of the race for the playoffs. The Dhoni-led team are scheduled to play two more matches as they remain at the ninth spot in the IPL rankings.
Five-time IPL winners Mumbai Indians, sitting 10th, are already out of the title race.