Handscomb ran himself out in his eagerness to help Warner reach the 100 mark when he sprinted for a single only to be sent back - a desperate dive not enough to beat Shakib Al Hasan's direct throw from square leg.
Warner brought up his 20th Test century with his fifth boundary, driving a Nasir Hossain full toss through cover.
The century followed Warner's 112 in the first test in Dhaka where Australia were beaten inside four days by the hosts.
Bangladesh, , took the new ball as soon as it was available and Mustafizur Rahman nearly tasted immediate success when he induced a lazy drive from Glenn Maxwell, but Mehidy Hasan spilled the catch at gully.
Mustafizur dismissed Warner for 123 with a short-pitched ball which the batsman hit to leg slip where Imrul Kayes fumbled before finally taking the catch.
Warner hit only seven boundaries in his cultured knock, painstakingly constructed over six hours.
Mehidy dropped Hilton Cartwright on six off his own bowling, the straight drive bursting through the spinner's hands and hitting him in the rib cage. Cartwright made 18 before edging the spinner to Soumya Sarkar in the slip.
Maxwell, who lived dangerously at the other end, made 38 while Matthew Wade managed eight. Ashton Agar made 22 down the order.
Bangladesh claimed four wickets in the 23-over final session to stay in the contest. Mustafizur, celebrating his 22nd birthday, claimed (3-84) and off-spinner Mehidy (3-93) shared six wickets.
David Warner led Australia's impressive reply on day two of the second Test against Bangladesh at Chittagong.
The opener struck 88 not out as the Aussies, looking to level the series after last week's surprise defeat in Mirpur, closed in on Bangladesh's first-innings total of 305.
Warner was ably assisted by captain Steve Smith, who looked well set for a possible 21st Test century until he played round a straight one from Taijul Islam on 58.
Peter Handscomb then continued where Smith had left off with an unbeaten 69 despite appearing to struggle with the heat as stumps approached.
Their stand of 127 ensured Australia closed on 225 for two to trail the hosts by 80 runs with eight wickets remaining.
Earlier, Bangladesh captain Mushfiqur Rahim added just six runs to his overnight 62 not out but Nasir Hossain hit 45 to help them past 300.
Spinner Nathan Lyon wrapped up the innings when he removed Taijul to finish with figures of seven for 94.
Bangladesh are after captain Mushfiqur Rahim and Sabbir Rahman dug in to frustrate the tourists on the opening day of the second Test in Chittagong.
The pair put on 105 for the sixth wicket after Australia, looking to level the series after last week's surprise loss in Dhaka, had made a fine start.
Nathan Lyon (five for 77) claimed four early wickets as Bangladesh slipped to 117 for five but Mushfiqur (62 not out) and Sabbir (66) lifted them to 253 for six at the close.
Australia put their faith in spinners, selecting three of them in Lyon, Steve O'Keefe and Ashton Agar, and just one specialist seamer in Pat Cummins.
Lyon took the new ball and made an immediate impact, trapping Tamim Iqbal, Imrul Kayes, Soumyar Sarkar and Mominul Haque leg before.
When Agar had Shakib Al Hasan caught behind by Matthew Wade, the tourists were well on top but they were unable to build on that advantage.
Mushfiqur held firm while Sabbir played fluently before Lyon, who bowled 28 of the day's 90 overs, had him stumped by Wade to become his fifth victim.
Mushfiqur will resume on day two alongside Nasir Hossain, who was unbeaten on 19.