Batting at No.3, the 22-year-old smashed 72 not out off 44 balls with five sixes and five fours as last year's finalists, coached by Aussie legend Ricky Ponting, survived a middle-order collapse to reach 7-165 with five balls to spare.
Fast bowler Vijayakumar Vyshak (3-34) and experienced spinner Yuzvendra Chahal (2-28) had earlier bowled well in the middle overs to restrict Gujarat to 6-162 after Punjab captain Shreyas Iyer won the toss and elected to field.
"I was looking forward to the experience of playing with some world-class players, but to put that performance in tonight, very happy," Connolly said.
"I'm just more of a tempo player, feel the flow of my hands, don't hit too many big sixes."
Punjab had the chase under control at 2-110 in 12 overs before impact substitute Prasidh Krishna rattled the middle order with three wickets in 10 balls and the home team slumped to 118-6 in the 15th over.
But Connolly didn't lose patience with wickets falling around him and provided Punjab a winning start to the tournament.
Krishna ignited the collapse when Iyer clipped a straightforward catch to mid-wicket of the first ball he faced after he was struck hard on his hand at the non-striker by a hard drive from Connolly and needed on-field treatment.
Shashank Singh was undone by a sharp short ball and got a faint edge behind the wicket while another Aussie, Marcus Stoinis, sliced a catch to Rashid Khan at third of another short delivery.
Marco Jansen smashed a six but got deceived by a slower delivery from Gujarat debutant Ashok Sharma and offered a tame catch at covers with Punjab still needing 19 runs for victory.
The third Aussie in the Punjab line-up, Xavier Bartlett (11 not out), eased the nerves when he pulled Krishna for a six over mid-wicket before Connolly guided Punjab to victory.
Earlier, skipper Shubman Gill top-scored with 39 off 27 balls, but none of the top-order Gujarat batters could convert starts into meaningful knocks.
Gill started off well with his fluent boundaries in the power play, but Gujarat, watched by new Aussie batting coach Matthew Hayden, got tied down after scoring 1-54 in the first six overs.
Gill dominated Chahal early on with his trademark sweep shots before he got caught at mid-wicket while trying another sweep against the spinner as Gujarat couldn't find momentum after power play.
Jos Buttler scored 38 off 33 balls, that included 14 dot balls before he holed out to long-on at the start of death overs.
Punjab did well in the last five overs and allowed Gujarat score only 34 runs with Jansen (1-20) intelligently varying his pace. Arshdeep Singh's (0-42) long 11-ball last over went for 12 runs that included a no-ball and four wide balls.
