Stoinis did it with bat and ball, helping Australia out of a tricky position following a sluggish start to their innings in Bridgetown.
The 34-year-old became just the third player in T20 World Cup history - after Dwayne Bravo (West Indies) and Shane Watson (Australia) - to score a 50 and take at least three wickets in a game.
After being sent in to bat by Oman captain Aqib Ilyas, Australia struggled early before reaching 5-164 in an unconvincing 20 overs at the crease.
Stoinis fired with an unbeaten 67 from 36 balls, smashing half a dozen sixes, including four during a vital 15th over.
Veteran opener David Warner and Stoinis put on 102 for the fourth wicket after finally adjusting to the spin-friendly pitch, having at one point gone 30 deliveries between boundaries.
David Warner has become Australia's highest T20I run-scorer after his World Cup 56 against Oman. (AP PHOTO)
Warner - who took his overall T20 total for Australia to 3155 runs - anchored the innings during his 51-ball stay, in the process overtaking his former opening partner Aaron Finch (3120) as Australia's most prolific batter in the format.
Oman never looked like chasing down the total, slumping to 4-34 in the eighth over as Stoinis claimed 3-19 to go with his brutal performance with the bat.
"It was a nice start," Stoinis said.
"Maybe a few cobwebs for us but we'll be better for the run.
"Davey and I talked about targeting the first couple of balls and when you hit a couple of balls for a boundary you think, 'I may as well try and turn this into a big over.'"
Marcus Stoinis was superb with both bat and ball! — cricket.com.au (@cricketcomau) #T20WorldCup pic.twitter.com/2E7warx4FwJune 6, 2024
Ayaan Khan (36) and Mehran Khan (27) showed some resistance for Oman, who lost to Namibia in their first match of the tournament, as they finished on 9-125.
In typical Mitchell Starc fashion, the left-arm quick took a wicket in his first over, trapping Pratik Athavale lbw for a duck.
But Starc (2-20) had to leave the field after his third over with soreness in his lower leg that Australia declared was only cramp.
Starc, who spent more than two days in transit en route to the Caribbean, tried to stretch on the field, but later came off for treatment and further assessment.
"There was no need to take unnecessary risks at this stage of the tournament," captain Mitch Marsh said.
"He said he's fine, just a cramp for the old boy."
Mitchell Starc had to leave the field after his third over with soreness in his lower leg. (AP PHOTO)
Earlier, star allrounder Glenn Maxwell, who has made five T20I centuries, was out first ball after Ilyas pulled off a flying catch at cover.
Pat Cummins, who led Sunrisers Hyderabad to the Indian Premier League final last month, was rested, with Australia instead bringing in Tasmanian pacer Nathan Ellis.
Australia are trying to become the first team to hold all three major International Cricket Council trophies - one-day international and T20 world cups as well as the World Test Championship - at the same time.
Australia's next game is a blockbuster against defending champions England at the same venue in Barbados on Saturday.
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