Lightning and heavy rain in Perth reduced Sunday's match to an 11-overs-a-side bash, with the Sixers posting 5-113 on the back of an unbeaten 46 off 21 balls from Jack Edwards.
Pakistan superstar Azam made just two - and he was lucky not to get out on two occasions before that - as the Sixers slumped to 4-41 before Edwards and Josh Philippe (28 off 18) launched a rescue mission.
In reply, the Scorchers reached their victory target in 10.1 overs on the back of Connolly's 59 off 31 balls, with his knock featuring five sixes and three fours.
"He's a matchwinner," Scorchers captain Ashton Turner said of Connolly.
"I mean, we're just seeing him get better and better every time he gets exposure to professional cricket. We've seen it for Australia as well.
"And clearly he's been identified as someone who's going to play a lot of international cricket across three formats, and we're just going to cherish the time that we've got him in orange.
"He's very much in the ilk of Shaun Marsh, who played some mightily good cricket for us for a long period of time.
"And he's hitting the ball into similar areas to Travis Head at times. It's really impressive."
There was drama an hour before the match when spectators were told to leave their seats and seek shelter as lightning started edging closer to the venue.
The stadium's lights flickered numerous times as the electrical storm gathered intensity.
Huge puddles formed on the turf as more than 15mm of rain fell within the space of an hour.
Play eventually began 80 minutes after the original start time, but it wasn't pretty early for the Sixers.
Azam was almost caught on two occasions before he even got off the mark - the first a flat-batted shot that just floated over Laurie Evans at mid-on, the second an aerial whack over mid-off that landed between two fielders.
His luck ran out next ball when he flat-batted paceman Brody Couch (2-12) down the ground for an easy catch.
Edwards cracked six fours and two sixes to guide Sydney to a competitive total.
Scorchers opener Mitch Marsh fell for a third-ball duck when Azam took a sharp catch at slip, and Finn Allen cracked 16 off eight balls before being snaffled in the deep.
Connolly, who shared the BBL player-of-the-season award with Glenn Maxwell last campaign, kept his cool and ensured the required run-rate never got out of hand.
He eventually holed out in the deep with just nine runs needed for victory, but the 26,823-strong crowd were already in party mode by that stage with the win effectively secured.
