Archer was named man of the match for his performance with the ball with figures 3-22 from his four overs as the Sunrisers won by three wickets.
Rajasthan restricted Mumbai to 167/7 after 20 overs, with Archer’s three wickets all coming in the 19th over.
Archer scored eight in the reply but Krishnappa Gowtham scored 33 from 11 balls and hit the winning runs with a six with two balls to spare. Sanju Samson top-scored with 52 and Ben Stokes scored 40 from 27 balls.
Archer could feature in Rajsthan’s next game on Sunday against the Sunrisers Hyderabad, where he could face Sussex team-mates Chris Jordan and Rashid Khan.
Meanwhile, Khan featured for the Sunrisers in their defeat to the Chennai Super Kings. The 19-year-old Afghan, who will play for Sussex in the Vitality T20 Blast this summer, produced figures of 1-49 from his four overs whilst Ambati Rayudu hit 79 from 37 balls to set a target of 182/3.
Even though New Zealand’s Kane Williamson top-scored with 84 from 51 balls, it wasn’t enough as they fell four runs short of the target.
The Sunrisers’ next game is away to the Mumbai Indians on Tuesday.