Finch blasted 106 from 114 balls, including nine fours and a six, to follow the 107 he made at the MCG on Sunday.
The burly opener looked like leading Australia to a score well past 300 but his dismissal, soon after that of Mitchell Marsh, ensured England were able to keep the Australian scoring under control.
Australia started their innings well with Finch and David Warner both dominating the English fast bowlers, who pitched too short on the Gabba wicket.
Moeen Ali had Warner caught at slip for 35, bringing Steve Smith to the crease.
More From This Section
The Australian captain struggled to 18 from 25 balls before part-time off-spinner Joe Root trapped him lbw.
Root then enticed Travis Head into charging down the wicket, only to chip a return catch to the bowler to leave the Australians on 124 for three.
Marsh and Finch consolidated but the scoring rate slowed as they struggled to get Ali, Root and leg-spinner Adil Rashid away.
The Australians then lost three wickets for nine runs to allow the English to take control.
Woakes then had Mitchell Starc caught in the deep before running out Andrew Tye on the last ball of the innings.