Under-pressure allrounder Cameron Green, who was bailed out by a stroke of luck, was looking composed on an unbeaten 43 at stumps on day three, along with wicketkeeper Alex Carey (19no), as Australia reached 4-161 in their second innings.
The hosts still trail by 67 runs after Bangladesh were dismissed for 426 just after lunch on Saturday.
Openers Jake Weatherald (0) and Travis Head (17) both chopped on to paceman Hasan Mahmud before tea.
After the break Marnus Labuschagne (31) played all around a half-volley from spinner Taijul Islam. For a set batsman, it was a cardinal error.
Green, on 12, inside-edged Taskin Ahmed for four via his leg stump in a moment of good fortune. Replays showed the ball hit the outside of the stump but the bails stayed on.
The dismissal of Smith (44) was out of character, as he pushed an innocuous delivery from spinner Mehidy Hasan Miraz straight back to the bowler.
Australia will have to rewrite the Test history books on home soil if they want to beat Bangladesh, after the visitors took a first-innings lead of 228.
Bangladesh allrounder Mehidy (65) and the tail took the total to 426 all out after lunch on Saturday.
Australia paceman Josh Hazlewood was the shining light with the ball, taking 6-89. The wicket of Mehidy was his 300th in Test cricket as he became the ninth Australian to reach the milestone.
The largest first-innings deficit Australia have come back from to win a Test on home soil was 206, when they trailed Pakistan at the SCG in 2010.
On that occasion Michael Hussey made an unbeaten 134 in the second innings and Australia bowled Pakistan out for 176 to win by 36 runs.
Hazlewood said the team would still press for victory.
"Once we tick those (67 runs) off we are still four down hopefully and we can start to build the pressure," he said.
"I'm not sure how the wicket will play, but it was showing signs of (playing) up and down.
"If we can bat deep into tomorrow and potentially the whole day there is always the opportunity to take 10 wickets on the final day.
"Winning is still in our mind and always is."
Australia's woes in the field were summed up before lunch on Saturday when No.10 batter Taskin hooked Pat Cummins for four with disdain. Next ball Smith dropped a sitter at deep backward square leg.
Head then spilled a skied pull shot off Mehidy, a difficult chance, to make it five dropped catches for the innings by Australia.
Mahmud (14) defied the bowlers for 92 deliveries after his brilliant 6-55 in the first innings in a show of batting application the Australians could well heed.
Spinner Taijul Islam (17) took a more aggressive approach, flipping Hazlewood over the square leg boundary before slashing the same bowler to gully, in a handy cameo.
Hazlewood vindicated his selection ahead of the unlucky Scott Boland and was the pick of the Aussie bowlers.
Bangladesh's first-innings lead was set up by a maiden ton for Tanzid Hasan (101) and an aggressive 84 by captain Najmul Hossain Shanto.
Australian legend Matthew Hayden told Channel Seven that a win was beyond the hosts after conceding such a big lead.
"You've got a completely different mindset as a batting unit," Hayden said at lunch.
"You are just looking to survive. That is it. You're not looking to win. It's survival. And that's a real struggle for Australian cricket. We do not like to be in that position."
