Having opted for a three-pronged spin attack in the first of the two test matches, Bangladesh captain Mushfiqur Rahim elected to bat but Cummins forced them on to the back foot immediately.
Australian cricket team is Bangladesh and preparing for the Test series.
Lyon then claimed the important wicket of Shakib with an extra bit of bounce from the track when the batsman edged him to Smith in the slips.
All-rounder Ashton Agar returned to Australian Test team after four years while the visitors also drafted in batsman Usman Khawaja.
Earlier, Shakib Al Hasan scored 84 and Tamim Iqbal 71 in their 50th test as Bangladesh recovered from paceman Pat Cummins onslaught to amass 260. He then removed Mehidy Hasan to get past Benaud to sit behind only Shane Warne (708) among Australia's all-time best spinners.
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Nathan Lyon will shoulder the responsibility of being the lead spinner with Agar and part-time option of Maxwell are the other options for Smith as they bid to take an early 1-0 lead in the two match series.
Both sides enjoyed spells of dominance in a fluctuating day that started brightly for Australia but quickly became a gripping contest.
On the turn: Australia's Glenn Maxwell, second from left, wrought the dismissal Bangladesh's Amim Iqbal. With Hasan being a right arm bowler, he could pick up early wickets easily.
Shafiul Islam and Mehedi Hasan opened bowling.
Cummins bowled back of a length to great effect to remove opener Soumya Sarkar (eight) with his fifth delivery Shar-e-Bangla National Cricket Stadium, the left-hander edging to Peter Handscomb in the gully.
Things looked even more ominous a ball later as wicketkeeper Matthew Wade took another catch, with Shabbir Rahman not adjusting to Cummins' late swing.
Shabbir's golden duck further riled those Bangladeshis who criticised the call to overlook Mominul Haque, whose Test batting average is more than 46.