New Zealand left-armer McClenaghan picked up 3-20 including the key wicket of Shikhar Dhawan (42) as the Indians successfully defended a total of 157-8 as the visitors slumped to 137-8, Lasith Malinga sealing the victory with three wickets in four balls in the penultimate over.
Mumbai, who went into the match with just one win in their opening six games and who remain bottom of the table, began positively after choosing to bat as Parthiv Patel (17) put on 42 for the first wicket with Lendl Simmons (51) in 5.4 overs before holing out off Dale Steyn (2-38).
Unmukt Chand fell cheaply but Simmons and Rohit Sharma laid the foundations for a challenging total with a third-wicket stand of 51.
Simmons struck six fours and one six off 42 balls only to give away his wicket with a swipe at Steyn, who knocked back middle stump.
Crucially Sharma???s cameo of 24 off 15 balls came to an end in the following over, caught in the deep off Karn Sharma, to leave Mumbai wobbling on 108-4.
With their momentum wavering, Kieron Pollard stepped in to hit 33 off 24 balls and drag Mumbai up to a modest total that looked around 20 runs light at the halfway stage.
That looked even more the case as Shikhar Dhawan struck 42 off just 29 balls to dominate a first-wicket stand of 45 in five overs with skipper David Warner (nine).
But the departure of the captain, caught at third man as he sought to upper cut Malinga, was followed swiftly by Dhawan???s downfall courtesy of a sharp catch at midwicket by Malinga off McClenaghan.
No batsman aside from Dhawan scored at faster than a run-a-ball in the innings and as McClenaghan and Malinga tightened the screw, the Sunrisers lost wickets at regular intervals.
Ravi Bopara struck just one four in his 27-ball 23 and with the game as good as up Malinga removed Hanumar Vihari, then sent back Praveen Kumar and Dale Steyn in successive balls to complete an impressive bowling display.