Spinner Nasum Ahmed claimed four wickets as Bangladesh defended 131 to beat Australia for the first time in Twenty20 internationals on Tuesday in Dhaka.
Nasum, a left-arm spinner, returned career-best figures of 4-19 to help dismiss the visitors for 108 as Bangladesh took a 1-0 lead in the five-match series.
Mustafizur Rahman and fellow left-arm quick Shoriful Islam took two wickets each to play key roles in their team's maiden T20 win over Australia in five attempts.
This is the first bilateral series between the two sides as Australia won their four matches against Bangladesh in T20 World Cups.
“We had a team talk during the halfway stage and said that we were 10 runs short, so we have to field well and put more effort,” skipper Mahmudullah Riyad said after the win.
“That's what you need while defending low totals and have attacking mindset. They are a very good side and we need to tick the boxes we failed today in the coming games.”
Mitchell Marsh made a valiant 45 but the away batting faltered from the start with Alex Carey bowled on the first ball to spinner Mahedi Hasan.
Marsh tried to get the chase on track with few partnerships including a 38-run fourth-wicket stand with skipper Matthew Wade, who made 13.
Mitchell Starc was the only other Australian batsman to manage double figures before he was bowled for 14 on the last ball of the match.
Earlier, pace bowler Josh Hazlewood returned figures of 3-24 in Bangladesh's 131-7 after Australia elected to bowl first. Starc took two wickets.
Shakib Al Hasan, who survived two dropped chances on 18 and 23, top-scored with 36 off 33 balls before being bowled by Hazlewood in the 17th over.
Opener Mohammad Naim (30), Mahmudullah (20) and Afif Hossain (23 not out) also contributed with the bat.
“I thought the bowlers did a terrific job to restrict them to 130,” said Wade.
“But having lost three for 10, me and Marsh tried to resurrect the innings but unfortunately we couldn't do that. We have to find ways to score runs on such pitches.”
The second T20 is on Wednesday at the same venue.