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.
Due to the vital partnership, Australia went on to post 377 for 9 wickets, leading by 72 runs in the first innings in a match curtailed due to rain.
Warner chalked up 123, having reached the milestone off 209 balls, in an uncharacteristically circumspect knock which included just seven boundaries.
Australia ended day two on 225/2 and will continue batting on Wednesday at 10:00 local time or 04:00 GMT.
These centuries will bolster Warner's reputation as a batsman who can make runs in toughest of conditions, and will also raise hopes of Australia's ultimate aim of winning a Test series in India when they travel for the Border-Gavaskar Trophy in 2021.
Lyon took the new ball and made an immediate impact, trapping Tamim Iqbal, Imrul Kayes, Soumyar Sarkar and Mominul Haque leg before.
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But with this innings, Warner will definitely be pleased as he managed to improve upon his indifferent sub-continental record.
David Warner was unbeaten on 88, delivering another big subcontinent score after making a century in the first Test in Dhaka.
No player was injured, Cricket Australia (CA) confirmed according to a espncricinfo report.
Pat Cummins (4) lasted just eight deliveries when he shouldered arms to Mehedy, but didn't expect it to turn a mile before hitting him on the pads without offering a shot. Soon after the Day-1 play wrapped up, the visiting team bus along with the players was on the way back to the hotel. Obviously, they spread the field quite a lot so it's hard to hit boundaries. "The way he has adapted and actually gone and batted in a different way in the last couple of innings has been spectacular for David".
Australia's Test ranking could slide to sixth if it loses the second Test. He has taken only 66 matches to bring up 20 Test match century.
"I'm doing myself and the team as many favours as I can by trying to put on runs, as everyone else is".
Australia took control of the second Test match against Bangladesh on Day 2 at Chittagong and will now look to drill home the advantage.