Perry picked up two wickets in her first five balls and finished with 2-14 from three overs as the Tigers were restricted to a paltry 8-77 from their 20 overs after the Aussies won the toss and elected to bowl at Leeds on Wednesday.
Australia capitalised on a favourable, green-tinged Headingley pitch, before strolling past the target with 63 deliveries to spare, making 1-78.
Perry wasn't among the six bowlers used in Australia's tournament opener against South Africa and had sent down only 14 overs in 36 matches since the last World Cup.
Australia's greatest allrounder looked comfortable and dangerous with ball in hand.
"Any opportunity to contribute, I really enjoy doing that with this group," Perry said.
"I've been chipping away with my bowling. I love doing it - it keeps me interested at training to have a trundle.
"To have an opportunity today was really enjoyable."
Georgia Voll (45 not out off 32 balls) hit six fours and a six, while player-of-the-match Perry (19no) capped a fine allround performance by hitting the winning runs.
Australia were forced to make two changes from the team that beat South Africa in the opening game, with Megan Schutt and Grace Harris coming in for Phoebe Litchfield and Ash Gardner.
Litchfield failed a fitness test on the morning of the game after suffering a quad injury while batting against the Proteas, while Gardner was unavailable with a sprained ankle.
The only blemish in the Aussies' clinical reply was the loss of Beth Mooney (10), who fell to an excellent catch by Ritu Moni at mid-off.
Earlier, captain Sophie Molineux (2-14), Perry (2-14), Kim Garth (2-18) and Annabel Sutherland (1-8 off three overs) blanketed the Tigers, who hit just six boundaries and no sixes.
In challenging batting conditions, Bangladesh were reduced to 5-27 in the eighth over, before skipper Nigar Sultana (27) and Ritu (16) - the only two to reach double figures - put on 32 for the sixth wicket to provide some respectability.
Their gritty partnership ended when Ritu lofted Georgia Wareham to Georgia Voll at midwicket, before Molineux trapped opposite number Nigar lbw.
Australia play Netherlands at Southampton's Rose Bowl on Saturday evening (AEST).
