Pat Cummins became only the second Australian to take a T20 World Cup hat-trick - after Brett Lee in 2007 - to help slow Bangladesh down late and restrict the underdogs to 8-140 in Antigua.
Winning the toss, Australia had Bangladesh's out-of-form batting order on the back foot when a classic full Mitch Starc (1-21) delivery sent Tanzid Hasan (0) back three balls into Friday's innings.
But after that, it was veteran tweaker Zampa (2-24) who did the heavy lifting, twice breaking through as Bangladesh lost 3-26 either side of drinks.
Player of the match in two of the four group-stage games, Zampa ended the 58-run partnership that had helped slow-starting Bangladesh recover from the early loss of Hasan.
Litton Das (16 off 25 balls) misplayed a sweep shot, Zampa's delivery bouncing from his knee onto the stumps, before the leg-spinner also snared Najmul Hossain Shanto (41 off 36).
Bangladesh's Litton Das loses his wicket to Adam Zampa in Antigua. (AP PHOTO)
The innings' anchor early on, captain Shanto did not even contemplate a review when Zampa struck his pad on the last ball of the 13th over.
With that dismissal, Zampa moved into outright second for most wickets at the tournament (11), trailing only Afghan quick Fazalhaq Farooqi (15).
The tournament's form allrounder, Marcus Stoinis (1-24) returned to the bowling attack after soreness kept him from taking the ball against Scotland in Australia's final group game.
He caught and bowled Shakib Al Hasan (eight) when the veteran skied his delivery, but copped some punishment as Towhid Hridoy (40 off 28) breathed life into Bangladesh's innings.
Cummins (3-29) saw to it that Bangladesh never truly clicked into gear, though, bowling Mahmudullah for two and then enticing Mahedi Hasan into finding Zampa at third man with the first ball he faced.
With his opening delivery of the 20th over, Cummins had his hat-trick when Hridoy clipped him to a stretching Josh Hazlewood in the cordon.
Cummins became only the fourth Australian to take a T20I hat-trick after Lee, Ashton Agar and Nathan Ellis, and the seventh player of any nation to achieve the feat at a T20 World Cup.
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