The South African came to the crease with Mumbai needing 34 runs off just 14 balls and Peterson plundered 16 not out off seven balls, with two fours and a six off successive deliveries.
Questions will be asked as to Peterson’s treatment by captain Harbhajan Singh, who bowled him for just one over, while he conceded 29 runs in three overs, and then batted before the left-hander but made a two-ball duck.
Rayudu blasted 34 not out off 17 balls, with a pair of fours and sixes, and hit the winning boundary as Mumbai won with a ball to spare.
The visitors needed 32 off the last two overs and Punjab will be ruing leg-spinner Piyush Chawla’s penultimate over of the innings, which cost an astonishing 27 runs.
Peterson, a clever cricketer who will always find a way to score, used a reverse-sweep and a switch-hit to hit the first two balls for four and then launched a slog-sweep for six.
Rayudu then hit the last two balls of the over for six, leaving Mumbai with just five runs to get off Azhar Mahmood’s final over and they did it in singles before the 26-year-old Indian prospect finished the match with a crunching cover drive.
The Kings XI probably thought they had the game in the bag after Mahmood and seamer Parvinder Awana had reduced Mumbai, chasing 169, to 135 for six in the 18th over.
Mumbai will be concerned that they collapsed after a good start, James Franklin (22) and Sachin Tendulkar (34) putting on 52 for the first wicket, before Rohit Sharma added a quickfire 50 off just 30 balls, with three sixes.
But the introduction of Pakistani paceman Mahmood brought a flurry of quick wickets as he had both openers caught behind by Nitin Saini.
The next over saw Chawla claim his one wicket as Dinesh Karthik (3) picked out Mahmood at short fine leg with a scoop shot.
Awana claimed the key wicket of Kieron Pollard for just three when the elevation of the West Indian’s flick over midwicket was just not enough and he was caught by Mandeep Singh.
Rohit then obviously became the key man as he played probably his best innings of the tournament before he slashed the nagging medium-pace of Awana to the sharp Mahmood at backward point.
The wicket of Harbhajan fell two balls later, but the skills of Peterson and Rayudu were enough to complete the victory.
The experienced Mahmood, who finished with two for 23, and Praveen Kumar, who conceded just 25 runs in his four overs, were the best of the Punjab bowlers.
David Hussey had earlier put his younger teammates to shame as he almost single-handedly steered the Kings XI to 168 for three.
Captain Hussey, the younger brother of Australian star Michael, slammed 68 not out off 40 balls to lift the home team to a competitive total on a pitch that was tough to score on when the ball got soft or the pace was taken off.
The 34-year-old made twice as many runs as any of his teammates, with South African David Miller’s 34 not out being the next highest score.
The multi-faceted Mumbai attack – seven bowlers were used – was initially pretty effective in stifling the run-rate and left-armer RP Singh was excellent up front, taking one for 27 in four overs.
Run-scoring was a little easier against the new ball and openers Saini and Mandeep put on 35 in the first five overs before Mandeep was caught behind off Singh for 22.
With Saini leaving the crease for 17 two overs later, his skied hook shot off paceman Clint McKay going to fine leg, the two Australians and Punjab’s most experienced batsmen – Shaun Marsh and Hussey – were brought together.
Marsh struggled to 17 off 26 balls before medium-pacer Franklin had him caught in an outstanding 13th over that cost just five runs, but Hussey batted through to the close.
Generally playing with a straight bat, Hussey punched his way to four fours and four sixes and was a cut above the other Kings XI batsmen.
But Miller served up a blistering cameo in the closing overs, lashing his 34 not out off just 17 balls, with two of his sixes coming off the last two balls of the innings, bowled by Munaf Patel, who the left-hander had also hit for six in his previous over.
Munaf had bowled his first two overs for just nine runs, including a maiden, but fell apart at the death as Hussey and Miller plundered 32 runs off his last 12 deliveries.
It said little for Harbhajan’s captaincy that spinner Peterson, who bowled the first over of the innings and conceded just three runs, was not used again, with the captain coming in for punishment as his three overs cost 29 runs.
Franklin also just bowled the single over.