Handscomb, who scored a half-century on debut against South Africa in Adelaide last month, brought up his maiden Test hundred with a glorious square drive to the boundary.
At tea, Australia were 377 for seven with Handscomb unbeaten on 103 and Josh Hazlewood on eight.
But it was a better session for the tourists, who fought back with four dismissals, despite yet more dropped catches.
The Australians, who resumed at 288 for three, lost skipper Steve Smith for 130, Nic Maddinson for one, Matthew Wade seven and Mitchell Starc 10 in the pre-tea session.
It ended a 172-run stand with Handscomb and he departed with his side in command at 323 for four.
Smith had some luck along the way with his 16th Test century, offering three chances during his 222-ball knock.
The Australian captain was dropped on 53 by Ahmed off spinner Azhar Ali and on 97 he survived in bizarre circumstances when a faint outside edge off Mohammad Amir was taken by Ahmed, but there was no appeal.
On 129, he advanced down the wicket to luckless leg-spinner Yasir Shah and skied to long-off, where Amir made a hash of the catch.
Maddinson was dropped by Azhar at short leg before scoring but was out soon after, caught behind off Wahab.
The tentative Maddinson lasted just eight balls, coming on top of his 12-ball duck on debut against South Africa in Adelaide last month to put his Test place in peril.
Wicketkeeper Wade fell to a poor shot, edging to third slip off Amir, and Starc hit out for 10 before Amir had him caught at first slip.
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