Opening batsman Shikhar Dhawan smashed an unbeaten 88, while all-rounder Rishi Dhawan (2-39) took the important wickets of Shivam Dube and Mahendra Singh Dhoni cheaply during the chase to seal victory despite a defiant 78 from 39 balls from Ambati Rayudu in Mumbai.
Dhawan, who was playing his 200th IPL match, hit nine fours and two sixes in his 59-ball innings to guide Punjab to 187-4, with Sri Lanka's Bhanuka Rajapaksa chipping in with 42 from 32.
Title holders Chennai slipped to 40-3 before Rayudu brought them back in the match with an innings that included six sixes and seven fours. But once he was bowled by Kagiso Rabada (2-23) in the 18th over, the game was as good as over.
Dhoni attempted to repeat his heroics of the previous match when he scored 16 off the final four balls against Mumbai but this time 27 runs from the 20th over bowled by domestic cricket veteran Rishi proved too much of a challenge.
Punjab co-owner and Bollywood star Preity Zinta was in the stands at the Wankhede Stadium to witness the win.
Medium-pacer Rishi also stood out - for the face guard he wore while bowling. The all-rounder was hit on the face during the domestic season and had to undergo surgery, and decided to protect himself from further injury.
"I think he [Rayudu] was batting brilliantly but we could have restricted them under 170-175," said Chennai captain Ravindra Jadeja.
"I think not getting a good start in the first six [overs], we're not putting runs on the board in the first six. So we need to improve on that and come back strong."
Punjab captain Mayank Agarwal credited his main seamers Rabada and Arshdeep Singh (1-23) for restricting the batsmen.
Singh gave away just eight runs to Dhoni and Jadeja in the penultimate over of the match, leaving 27 to be made off the last six balls.
"I thought Arsh bowled exceedingly well. I must credit him. Throughout the season, he's stood up in tough moments and he's bowled the tough overs," Agarwal said.
"He's always put his hand up and said, 'Gimme the ball! So he's been great for us. Even Rabada was fantastic, to get Rayudu at that time was very crucial."