Australia will resume at 2-225 today, having fought bravely in stifling heat after dismissing Bangladesh for 305.
With Australia trailing 1-0 in the series, Handscomb and Warner put on an inspired 127-run partnership that could go a long way towards helping the visitors square the ledger.
If the Australians can bat throughout most of day three, they will fancy themselves to restrict Bangladesh’s second-innings total on a deteriorating pitch.
Handscomb received medical treatment but played through yesterday’s last session to finish 69 not out, while Warner was unbeaten on 88 after making a century in the first Test in Dhaka.
Play was paused for about five minutes late in the final session when Handscomb pulled up on his haunches, looking clearly unwell after running between the wickets.
The 26-year-old sat on the ground and was examined by team doctor Geoff Verrall, who applied cold towels to his head and neck.
Handscomb, who showed sublime footwork during his stylish knock, faced 113 deliveries in the difficult conditions, while the ultra-fit Warner dealt with 170.
“Really gutsy, Obviously it was pretty hot out there, and we saw that yesterday,” Australian coach Darren Lehmann said after play.
“You have to work really hard for your runs so, to get through in the way they played, was excellent.
“They’re in the ice baths now. We’ll leave it with the medical team to get them right for tomorrow but a very special day and, hopefully, tomorrow they can kick on.”
After scoring an aggressive ton in Dhaka, Warner changed gears with a patient and composed knock to bring up his 25th Test half-century from 98 balls.
The left-hander was dropped on 52, with Mominul Haque putting down a difficult chance at short-leg off the bowling of Taijul Islam.
He also benefited from a bungled stumping on 73, with Bangladesh skipper Mushfiqur Rahim fumbling the ball when Warner danced down the pitch and missed an attempted slog.
Mushfiqur, who top-scored for Bangladesh with 68, had earlier taken a spectacular diving leg-side catch off the bowling of paceman Mustafizur Rahman to dismiss Matt Renshaw (4).
Australian skipper Steve Smith made an imperious 58 before being clean-bowled by a straight delivery from left-arm spinner Taijul shortly before tea.
Smith fumed as he walked from the ground after leaving an inviting gap between bat and pad for Taijul to exploit in his first ball of the innings.
Nathan Lyon earlier finished with 7-94 – his fourth seven-wicket haul and the equal-second best figures of his career after claiming 8-50 against India earlier this year in Bangalore.
Lehmann said the offspinner just keeps getting better after another heroic subcontinent performance from the 29-year-old, who took his career Test wicket tally in Asia to 77, overtaking Glenn McGrath to sit second behind Shane Warne among Australian bowlers in the region.
Lyon’s performance on the subcontinent had been cause for consternation just 12 months ago, with Lehmann singling him out for criticism during Australia’s disastrous series whitewash in Sri Lanka.
But Lyon’s latest triumph, backing up the nine wickets he took in the first Test in Dhaka, underscored the fact that he has become one of Australia’s best performers on the subcontinent.
“It was fantastic. Obviously he’s just getting better and better in these conditions,” Lehmann said.
“He’s really growing as a leader in our pack, if you like, in these conditions.
“He’s made subtle changes – I think he said ‘bowl ugly’ – and that’s what he’s been prepared to do.
“I’ve been really pleased with his consistency and the way he’s changed.”
Lyon on Monday said he had studied footage of Indian offspin maestro Ravichandran Ashwin in a bid to improve his performance on the difficult subcontinent wickets.
“I’ve watched a lot of footage of Ashwin, who in my book is probably the No. 1 spinner in the world,” Lyon said.
“In these conditions, it’s all about adapting and learning… I have to put my ego away and really bowl with confidence, in my terms ‘bowl ugly.’
“That might be round arm for me, trying to undercut the ball a little bit more but using my stock ball as a variation.
“I think I’ve learnt a lot in the subcontinent and I’m embracing the challenge.”
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