Playing just his second Test, Overton removed the dangerous David Warner and local batsman Cameron Bancroft in quick succession to lift English spirits after the visitors suffered a dramatic batting collapse before lunch.
Australia were 88 for two at tea, with Steve Smith on 24 and Usman Khawaja on 14, in reply to England's total of 403, which looked respectable but was far short of what might have been expected when they reached 368 for four in the morning session.
After England's tail collapsed, things looked grim for the tourists immediately after lunch as Warner and Bancroft made a positive start to the Australian innings.
However, the introduction of Overton into the attack paid immediate dividends, with Warner caught behind for 22 to end a 44-run opening partnership.
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Overton nearly had a second wicket two balls later, when he dropped a tough caught-and-bowled chance from new batsman Khawaja.
He then darted a ball into the pads of Bancroft, and although on-field umpire Marais Erasmus turned down the initial appeal, it was given out by third umpire Aleem Dar on review.
England lost their last six wickets for just 35 runs in a spectacular collapse, having won the toss and elected to bat.
Bairstow put a controversial few weeks behind him on and off the field to post his fourth Test century as part of a 237-run stand with Malan, who notched his maiden Test century.
The pair put on a new record fifth-wicket partnership for England against Australia, beating a long-standing mark set in 1938 by Eddie Paynter and Denis Compton at Trent Bridge.
Malan's wicket was the first to fall Friday, with his breakthrough innings finally coming to an end when he was brilliantly caught by substitute fielder Peter Handscomb from the bowling of Nathan Lyon.
Promoted to six, Bairstow was bowled by Mitchell Starc as wickets fell in a rush, with Starc finishing as the pick of the bowlers with four in total.