Schutt snared 3-12 off four overs to restrict Sri Lanka to 7-93 at the Sharjah Cricket Stadium in the United Arab Emirates on Saturday.
In reply, Australia slumped to 2-14 and then 3-35 as captain Alyssa Healy (4), Georgia Wareham (3) and Ellyse Perry (17) all departed.
Beth Mooney guided Australia to victory with her unbeaten 43. (AP PHOTO)
But Mooney helped settle the ship with an unbeaten 43 off 38 balls, guiding Australia to victory with 34 deliveries to spare.
"We were decent without being perfect," Healy said.Â
"We adapted to the conditions really quickly with the ball.
"To start the tournament 1-0 is a good result for us. Some confidence out of that."
After winning the toss and electing to bat, Sri Lanka slumped to 2-6 in the fourth over and never recovered as Australia's star-studded attack turned the screws.
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Spinner Sophie Molineux finished with 2-20 from her four overs, but it was Schutt who proved the trickiest to handle.
Schutt snared the scalp of opener Vishmi Gunaratne for a duck in the third over to kick-start Australia's dominant display.
The 31-year-old claimed another two wickets in the final over, lifting her overall wicket tally in T20 World Cups to 43.
That equals the mark set by South African Shabnim Ismail, who retired from international cricket last year.Â
The temperature soared into the mid-30s during the match, but Australia's use of daily sauna sessions last month meant they were well prepared for what was to come.
"I think we've finally acclimatised, because I didn't feel like I was going to die out there today," Schutt said with a laugh.
"I think Mooney lost about 16kg out there. We've played in hot conditions before. We play a summer sport.Â
"It is what it is. You've got to suck it up and get on with it. The breeze out here made it a lot easier today."
Nilakshi de Silva top-scored for Sri Lanka with an unbeaten 29 off 40 balls, while Harshitha Samarawickrama (23 off 35 balls) and Anushka Sanjeewani (16) were the only other batters to make it into double figures.
Australia captain Alyssa Healy appeals successfully for the wicket of Sri Lanka's Kavisha Dilhari. (AP PHOTO)
Australia opted against playing a fourth spinner, instead selecting fast bowler Darcie Brown.
The 21-year-old endured a tough outing, bowling three no-balls and a wide in her first and only over.
That meant that after just one legal delivery, Brown had figures of 0-10.
Brown finished with 0-12, and also dropped a simple catch in the field.
Georgia Wareham (1-21), Molineux and Ashleigh Gardner (1-14) were picked as Australia's spinning trio, and they all performed well.
Australia's run chase started in shaky fashion, with Healy clean bowled and Wareham run out via a brilliant direct hit from Kavisha Dilhari.
Mooney clearly found the heat to be a struggle but still ensured the early loss of wickets would only be a fleeting worry, producing a solid knock to help get Australia out of trouble.
The 30-year-old was gifted a life on 22 when she was dropped in the deep, but apart from that she seemed in complete control.Â
Australia, who are aiming for a fourth consecutive T20 World Cup crown, will be back in action on Tuesday when they take on New Zealand.
Sri Lanka, who are now 0-2 after losing to Pakistan and Australia, will be fighting to stay alive when they take on India on Wednesday.
