Bangladesh pace bowler Hasan Mahmud starred with 6-55 as Australia were bundled out for 198 in 53 overs at Marrara Stadium.
It was the 26-year-old's best Test figures in an innings.
Under pressure batters Jake Weatherald, Marnus Labuschagne and Cameron Green all failed.
In reply, the visitors went on the attack to finish the day at 1-96 with Tanzid Hasan (32 not out) and Mominul Haque (35no) relatively untroubled.
Mitchell Starc ended a breezy knock by Shadman Islam (20) to take his Test wicket tally to 434, the most ever by a left-arm bowler.
Smith had a few nervous moments before top-scoring with 71 after captain Pat Cummins won the toss.
The veteran was out skying a swat off Mahmud chasing runs with the tail.Â
Smith is yet to score a Test century against Bangladesh, the only one of the nine Test-playing nations he has played against where he has not achieved the milestone.
The rest of the Australian batters struggled as Bangladesh pace trio Taskin Ahmed, Ebadot Hossain and Mahmud bowled superbly on a surface that assisted the seam bowlers.
Mahmud said the bowling group was thrilled with Cummins' call to bat first.
"It was an opportunity I was thinking was very good for our bowlers, particularly in these conditions," he said.
"The way we started was awesome and we just kept bowling in our channels."
Ebadot (2-39) should have claimed Smith for two, but he was dropped by Tanzid Hasan at third slip.
Bangladesh then lost a second review when Smith was given not out when on seven after Ebadot was convinced he'd nicked one.
Smith admitted after play it was a poor day for the hosts.
"Not our best," he said.
"I actually forgot how to hold a bat for the first 30 balls, then I figured it out and felt really good.
"There's still a bit of cricket to be played ... so hopefully we start well in the morning, take a few early (wickets) and put a bit of pressure on."
There was a sense Australia's No.4 had made a move towards the pavilion, but Snicko showed nothing on his outside edge.
Ebadot cleaned up Alex Carey (19) after lunch and Taskin castled Beau Webster (12) with an absolute beauty. Pat Cummins (nine) and Starc (one) both nicked off to Mahmud before tea.
Smith made a statement when he launched veteran spinner Taijal Islam over the ropes and then through the covers in his first over of left-arm spin.
Earlier, the battles of Weatherald, Labuschagne and Green in Test cricket continued as Bangladesh struck early blows to have the hosts 4-74 at lunch.
Weatherald, who has passed 50 just once in 11 Test innings, had an opportunity to enhance his chances of staying in the top order in front of a partisan crowd in the city of his birth, but was unable to make the most of a start.
It was no surprise when the 31-year-old edged impressive paceman Mahmud on the cusp of drinks to bring an end to a scratchy 23 off 37 deliveries.
Opening partner Travis Head (22) followed soon after, chopping on to Mahmud when he went around the wicket.
Ebadot had Labuschagne (one) edging to captain Najmul Hossain Shanto with an indecisive poke as his drought without scoring a Test ton extended to 37 innings.
A horror shot by allrounder Cameron Green (13) on the stroke of lunch went straight to midwicket, another misadventure by a player who is yet to turn his promise into consistent runs.
The Test is the first played at the venue since 2004 and just the third in Darwin history after Bangladesh were the first visiting side in 2003.
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