Steve Smith hit another Ashes century, his second of the match, to leave England facing a huge battle to save the first Test.
It is Smith’s first test since returning from a ban for his part in the ball-tampering scandal.
Smith added 142 runs to his first-innings 144.
Matthew Wade also scored 110, with Australia taking their overnight 124-3 to 487-7.
They have declared setting England 398 to win.
England openers Rory Burns and Jason Roy saw out the last seven overs of the day to close on 13-0.
England will have to bat out the final day if they are to avoid losing an Edgbaston Test for the first time in 11 years.