Asked to bat first, efforts from Shakib Al Hasan (36) and Mohammad Naim (30) pushed the Tigers to 131 for seven at the Sher-e-Bangla National Cricket Stadium on Tuesday evening.
Nasum then went through the Aussie batting line-up with his slow left-arm orthodox, picking up 4-19 in the four-over quota as Bangladesh took a 1-0 lead in the five-match series.
Seamers Mustafizur Rahman and Shoriful Islam chipped in with two wickets each to bundle out the tourists for 108.
The only Aussie resistance came from Mitchell Marsh who scored a run-a-ball 45 before falling to Nasum.
In pursuit of a modest 132, Australia had a stuttering start when Mahedi Hasan went through the gates of opener Alex Carey with the first ball of the innings.
Nasum then took out Josh Philippe (9), stumped when danced down the track. Shakib sent Moises Henriques (1) packing in the third over to leave Australia reeling on 11 for three.
Marsh then steadied the ship with skipper Matthew Wade (13). The duo patiently worked the spinners around running hard between the wickets and picking up occasional boundaries.
The track offered turn for the spinners while the seamers met low-bounce on the pitch.
In trying to break free of Bangladeshi spinners who displayed much control and confidence, Wade went after a wayward one from Nasum but ended up sweeping to short-fine leg where Mustafizur pouched it safely.
The visitors plunged further into trouble when Ashton Agar (7) went back hit wicket off Nasum before Marsh and Ashton Turner (10) sliced Mustafizur to extra cover, as Marsh kept running out of partners.
Nasum, who ended with a career-best haul, bagged Marsh’s scalp for his fourth to cap off a brilliant performance with Australia still 47 runs shy and less than five overs remaining.
Australia tottered on into the final over but Mustafizur made sure there was no drama as he went through the gates of Mitchell Starc (14) for the final wicket with the last ball of the innings.
Earlier, Australia had the better of the opening stages of the game despite two sixes from Naim with Josh Hazlewood (3-24) leading the charge.
The Tigers looked uncomfortable from the start as well when Naim almost chopped on to the wicket, which his partner Soumya Sarkar (2) did in the fourth over off Hazlewood.
Naim showed promise with two more boundaries but a premeditated reverse sweep off Adam Zampa brought his downfall when he missed and was embarrassingly castled after taking his team past the powerplay with Shakib.
Skipper Mahmudullah (20) and Shakib then attempted to push the run rate with their skills against spin. But Zampa and Agar kept things tight, not allowing the batsmen to free their arms and keeping the run rate below six.
As the batsmen looked to change gears, Hazlewood took out both of them as Bangladesh struggled to launch a burst in the final overs.
Reduced to 104 for five with three overs remaining, a crucial cameo from Afif Hossain (23 off 17) pushed Bangladesh to the modest total, which ultimately proved too much for Australia to chase down.
The 26-year-old southpaw Nasum was adjudged Man of the Match.
At the presentation ceremony, Mahmudullah said: “We had a team talk that we were 10 runs short, so we have to field well. It was outstanding, the hunger and energy was there. The bowlers executed their plans really well.”
“You need to have an attacking mindset with the ball early on, that's what we did. I'm not over the top, we've won, it's gone. Now it's time to focus on the next game and try and be on top from the first ball, try and manage ourselves, keep our feet on the ground."