Victory means Ricky Ponting's Punjab team, beaten finalists in 2025, have won both their opening fixtures and are top, for the moment at least, of the nascent IPL table.
There was further Aussie assistance from Marcus Stoinis, there at the end with an unbeaten nine, including the winning boundary, that helped see Punjab home, and Xavier Bartlett who had earlier taken the big wicket of Chennai opener Sanju Samson in the second over.
Opening batters Priyansh Arya and Prabhsimran Singh gave Punjab a flying start to a target of 210 as they overhauled Chennai Super Kings on Friday.
Impact substitute Arya set the tone with 39 runs off 11 balls as he and Singh rumbled to 1-68 in the powerplay.
Captain Shreyas Iyer's 26-ball half-century and Connolly's 36 ensured Punjab reached 5-210 with eight balls to spare and won by five wickets.
Chennai thought they had put up a defendable total of 5-209 in front of their home supporters thanks to Ayush Mhatre's 73 off 43, Shivam Dube's unbeaten 45 off 27 and Sarfaraz Khan's cameo 32 off 12.
But Chennai's bowling was ordinary. Only five bowlers were used while allrounders Dube and debutant Prashant Veer were not used.
Iyer said: "That was an exceptional start for us. I feel the way they (Arya and Singh) have been batting has been phenomenal and it stabilises the rhythm for us.I am glad everyone is getting to bat. It gives immense confidence to the team."
Arya smacked fast bowler Matt Henry for three fours and a six in a 20-run second over after the left-hander hit Khaleel Ahmed for a four and a six off the first two legitimate balls in the first over.
Singh raised Punjab's 50 in only the third over when he took three boundaries off Anshul Kamboj.
Henry rattled Arya's off stump and Singh was run out in a mixup with Connolly when the Australian refused to go for a tight second run. Connolly holed out at long-on then Iyer took charge of the chase.
Iyer smashed three sixes and four boundaries in a 59-run stand with Nehal Wadhera that sealed the result.
Earlier, Iyer continued the template of teams preferring to chase when he won the toss and elected to field.
Mhatre showed plenty of aggression in a stand of 96 with captain Ruturaj Gaikwad, who made a scratchy 28 before falling to the IPL's leading wicket-taker, Yuzvendra Chahal, in the 12th over.
Mhatre looked set for a big knock after crashing five sixes and six boundaries and was livid with himself when he was caught at short third while attempting an extravagant shot against Vijakumar Vyshak (2-38).
Khan and Dube propelled Chennai beyond 200 but their bowlers could not tie down Punjab.
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