Rajasthan's victory on Sunday also knocked out the Ricky Ponting-coached Punjab Kings, who had occupied the fourth and last play-off spot.
Archer struck 32 off 15 balls to help Rajasthan score 8-205-in their must-win final league game, and then took 3-17 in four overs as Mumbai were restricted to 9-175.
"I don't think it's my best year, I have bowled a little bit better than this," Archer said after he was named player of the match.
"Every time you take the ball, you just try to bowl in good areas. Some days you're not going to get rewarded, happy that we got rewarded on the day we needed to win.
"I wasn't expecting to bat so early. Because of the impact sub, I hardly get to bat. When the coach said, 'Put the pads on,' I was surprised. I'm glad I was able to contribute. I class myself an all-rounder."
It meant Rajasthan (16 points) secured fourth spot — a point ahead of Punjab — with their eighth win in 14 games.
Mumbai finished ninth with 10 losses in 14 games, just above Lucknow on net run-rate.
Put into bat, Rajasthan lost 15-year-old Vaibhav Sooryavanshi for four but Yashasvi Jaiswal provided a quick start with 27 off 17 balls, including three sixes.
Rajasthan scored 3-54 in the powerplay – Jaiswal was caught off Will Jacks in the fourth over, with Sooryavanshi out four balls later to Deepak Chahar. Skipper Riyan Parag (14) was caught on the final ball of the powerplay.
Dhruv Jurel (38) and Dasun Shanaka (29) added 45 off 27 balls. Shanaka was run out just past the halfway stage.
Rajasthan slipped to 6-139 in 15.3 overs, with Jurel and Donovan Ferreira (18) losing their wickets.
Archer came to the rescue. He hit three sixes in his 15-ball innings and Ravindra Jadeja scored 19 not out off 11 as Rajasthan eventually crossed 200.
Archer then struck with the ball – Rohit Sharma was caught behind for a duck and Naman Dhir was bowled for six with Mumbai down to 2-18 in 2.3 overs.
Nandre Burger sent back Ryan Rickelton for 12 in the next over and Brijesh Sharma bowled Tilak Varma for three. Mumbai were 4-38 in 5.1 overs and finished the powerplay with 49-4.
Suryakumar Yadav scored 60 off 42 balls – only his second half-century this season – and rebuilt the chase, first with Jacks (33) and then skipper Hardik Pandya (34).
Yadav and Jacks added 63 off 38 balls, and then another 48 off 25 balls with Pandya, as Mumbai fought back.
But Archer returned to have Pandya caught in the 16th over and Mumbai's hopes faded.
Yadav was out caught in the 18th over as Mumbai's disappointing season approached its end. Burger finished with 2-43, while Sharma took 2-26 and wrist spinner Yash Raj Punja 2-44.
After Rajasthan's qualification, already eliminated Delhi Capitals beat Cameron Green's Kolkata Knight Riders by 40 runs in the season's final league game.
Lokesh Rahul hit 60 off 30 balls, with four sixes, as Delhi scored 203-5.
Skipper Ajinkya Rahane led Kolkata's chase with 63 off 39 balls, before wrist spinner Kuldeep Yadav turned the game with 3-29 while Green's fellow Australia star Mitchell Starc chipped in with 2-26 from three overs.
Kolkata collapsed from 3-128 to 163 all out in the span of 33 balls.
Kolkata (13 points) finished seventh in the table, with six wins in 14 games. Delhi (14 points) ended sixth with seven wins.
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