After winning the toss and electing to bat on Saturday, Sri Lanka slumped to 2-6 in the fourth over and never recovered as Australia's star-studded attack turned the screws.
Spinner Sophie Molineux finished with 2-20 from her four overs, but it was Schutt (3-12 off four overs) 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.
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 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 have won the T20 World Cup six times, and are aiming for four crowns in a row.
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