The tournament hosts started the game bruised by a crushing defeat against New Zealand in their first Super 12 match, needing a healthy win to repair both their pride and their net run-rate.
They were making heavy weather of the 158-run chase until Stoinis unleashed his display of ball-striking brilliance, crashing 59 not out in 18 game-changing deliveries.
The burly all-rounder had walked to the crease with Australia 89-3, needing 69 off 46 balls, and walked off again with the points in the bank and 3.3 overs unused.
He struck six sixes and four fours, casting the struggles of skipper Aaron Finch into stark relief.
Finch finished unbeaten on 31 and even scored the winning runs but his 42-ball stay proved a drag on Australia's progress and he was visibly yelling at his own lack of timing at one stage.
Seeing Australia prevail on their own soil is not a new experience for Sri Lanka head coach Chris Silverwood, who was sacked by England earlier this year after a humbling Ashes, but his old side still have a chance to deal a major blow when the old rivals meet on Friday.
The away side posted 157-6 after Finch has won the toss and opted to bowl first, recovering well from Pat Cummins' dismissal of key opener Kusal Mendis (5) in the second over.
Pathum Nissanka ground out 40 off 45 balls, with Charith Asalanka livelier for his unbeaten 38 off 25 deliveries.
Sri Lanka suffered bad luck when seamer Binura Fernando was forced off injured after sending down just five balls of the reply, while highly-rated spinner Wanindu Hasaranga had a day to forget. He shipped 53 from his three overs and leaked five sixes.
Finch's slow going, David Warner's (11) dismissal by Maheesh Theekshana and a smart catch to dismiss the aggressive Glenn Maxwell (23 off 12 balls) gave Sri Lanka a chance to take it all the way but Stoinis tore it away from them.
He arrived halfway through the 13th over, swung with gleeful abandon and reach his half-century in 17 balls, a new Australian record in the shortest format.