After winning the toss and electing to bat, Bangladesh were bundled out for an under-par 131 in 19 overs, which the Aussies moved past with 10 deliveries to spare at Chattogram on Wednesday to take a 1-0 series lead.
Cooper Connolly, the hero of Australia's ODI victory in Dhaka on Sunday, continued his form surge, top-scoring with 47 in an otherwise unconvincing chase from the tourists.
"Coops has had a taste of international cricket over the last 12 months and he's only going to continue to grow and get better the more comfortable he feels," said Australian captain Mitch Marsh, who missed the 2-1 ODI series loss with an ankle injury.
"It was great to see him bat like that on a reasonably tricky batting wicket.
"It's always nice to start a series with a win."
The Tigers made a fast start before recklessly submitting to Australia's slow bowlers, nine of the 10 wickets falling to spin.
Bowling allrounder Mahedi Hasan (29no), opener Saif Hassan (20) and No.3 Soumya Sarkar (17) were the only Bangladesh batters to pass 10.
Left-arm orthodox tweaker Davies (3-17) and leg-spinner Adam Zampa (3-18) were the chief destroyers.
Nikhil Chaudhary, Australia's other debutant, claimed 1-14 and took two catches, the first one a superb diving effort off Zampa at deep cover to dismiss Soumya.
Davies removed wicketkeeper Parvez Hossain Emon (10) with just his second international delivery, before Zampa became the first Aussie to take 150 wickets in men's T20s when Shoriful Islam picked out Tim David.
"It feels good - it means I've been around for a long time," Zampa said of his milestone.
"Obviously a proud achievement, especially in a game like this.
"We've had a tough time of it in the ODIs and to have a couple of young guys come in, Joel Davies and Nikhil Chaudhary, feel like they're at home in the team and look like they're a part of it, it feels good."
In reply, Josh Inglis (5) and returning captain Mitch Marsh (13) departed early in the chase, before Connolly steadied the ship.
The 23-year-old left-hander struck three sixes in his 27-ball knock and had a maiden T20 half-century in his sights when he holed out to Shoriful off first-gamer Abdul Gaffar Saqlain.
David (20), Chaudhary (18) and Matt Renshaw (18) fell with victory in sight, before Xavier Bartlett finished the Tigers off with a first-ball boundary to reach 6-133.
The second T20 will also be played at Chattogram on Friday.
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