Rory Burns added only eight runs to his overnight score before departing for 133, one of four wickets to fall on Saturday morning as the hosts slipped from 282-4 to 300-8.
England had nevertheless moved past Australia's first-innings total, and a 65-run stand between Woakes (37 not out) and Broad for the ninth wicket added to the lead before the hosts were all out for 374 when an early tea was taken.
When Broad was out for 29 from 67 deliveries, succumbing to Pat Cummins' short-ball tactic, Woakes was left with Jimmy Anderson for company.
England's record wicket-taker, moving gingerly after suffering a calf complaint earlier in the test, contributed three before spooning Lyon into the leg-side, ending a labored final wicket stand during which Woakes turned down a number of singles.
While Anderson was able to bat, it remains unclear whether he will bowl as he did not take to the field for the start of Australia's second innings.
England's morning wobble saw Burns, Ben Stokes, Moeen Ali and Jonny Bairstow also dismissed in quick succession. Stokes was the only batsman to show any fluency, reaching 50 before he aimed a cut at Cummins and nicked through to Tim Paine.
After more than seven hours in the middle, Burns was undone by Nathan Lyon's off spin, edging to Paine who held on instinctively.
Moeen went for a five-ball duck, and Bairstow managed only eight before steering Peter Siddle to David Warner at slip.