Medium pacer Marcus Stoinis led the way with four wickets as a disciplined Kings XI Punjab restricted Mumbai Indians to a modest 124 for nine in the stipulated 20 overs.
Stoinis returned brilliant figures of 4/15 in his full quota of four overs after Mumbai Indians skipper Rohit Sharma elected to bat.
In reply, Kings XI Punjab chased down the target of 125 with as many as three overs to spare.
Walking out to bat after the departure of Hashim Amla, Saha put on a match-winning partnership of 116 runs for the second wicket with Vijay to help KXIP register their fourth win in 11 outings. The duo consumed only 14.2 overs to stitch together the century stand.
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Saha reached his half century off just 38 balls, lofting Jasprit Bumrah over cover for a boundary. Vijay followed suit, as he too reached his fifty with a cut through the point region, an overthrow helping him to complete the second run and reach the landmark.
Saha struck six boundaries and a six in his strokeful knock, while Vijay hit five fours and cleared the fence.
While MI stayed at the fifth position with 12 points, Kings XI Punjab moved a step up to be in seventh place with eight points.
Earlier, Mumbai Indians were off to a disastrous start, losing two wickets for just eight runs on the board within the third over of the innings.
Chand failed to open his account at the Dr. Y S Rajasekhara Reddy ACA-VDCA Cricket Stadium. Ambabti Rayudu was done in by a delivery that swung in prodigiously and hit the off and middle stumps. Rayudu, too, was dismissed for nought, the bowler being seamer Sandeep Sharma.
Mumbai Indians were dealt a body blow when they lost the in-form Rohit, bowled by left-arm spinner Axar Patel. Rohit made 15 off 24 balls with a lone boundary.
The reigning champions sank deeper as they lost Nitish Rana (25) followed by the big wicket of Jos Buttler (9). Stoinis got both the wickets to leave Mumbai reeling at 61 for five in the 14th over. Rana struck three sixes while facing 28 balls.
Pollard was well supported by Krunal Pandya as he also, on and off, found the fence before holing out at the deep to end a sixth-wicket partnership of 42 that came off just 19 balls.
Stoinis then returned to dismantle Pollard's off stump and again pegged MI back at a crucial juncture. While Pollard smashed 27 off 20 balls with the help of three sixes, Pandya contributed a 12-ball 19, hitting a boundary and six.
Stoinis was supported by Sandeep Sharma (2/11) and Mohit (2/26).