Ponting was unbeaten on 31 from 42 balls including six boundaries while opener Hayden was 34 with the Australian score reading 74 for one.
Inspired by their brave batting, Bangladesh also showed plenty of spirit in the field and soon had Justin Langer caught at gully by Javed Omar off the bowling of Mashrafe Mortaza for one with the score on 14.
However, Australia regained their composure after Hayden had survived an edge to slip which didn't carry and a confident lbw appeal.
Bangladesh had resumed on their overnight score of 289 for eight and lasted just 13 balls on the second day's play as Jason Gillespie (3-57) and Glenn McGrath (1-57) dismissed tailenders Mashrafe Mortaza for eight and Tapash Baisya for 25.
Bangladesh scored their highest first innings total since their 400 on Test debut against India in Dhaka three years ago.
The score held special significance for the Asian nation after they were sent in to bat against the top-ranked team in the game on Australia's home turf.
Opener Hannan Sarkar, 20, made his highest Test score of 76 on Friday, sharing a 108-run stand for the second wicket with Habibul Bashar (46) while Sanwar Hossain (46) and wicketkeeper Khaled Mashud also took the fight to the Aussies.
Leg-spinner Stuart MacGill took 5-77 for Australia after the pace attack had struggled to break through the top order, with Brett Lee in particular proving expensive at 1-88 from 18 overs.
Australia lead the two-match series 1-0 after Bangladesh, who have not won a Test in 20 attempts, lost the first Test inside three days by an innings and 132 runs in Darwin last week.
