Rain is also forecast for Sydney tomorrow which may assist England's cause in playing for a draw. Earlier, spinner Jack Leach took four wickets for England, including Green and Alex Carey off the final two balls of Australia's innings, in what was his best bowling performance of the series.
He also dismissed Marcus Harris (27) and then got the prize wicket of Steve Smith for 23, bowling the star batsman with a delivery that kept low of the SCG pitch to reduce Australia to a potentially vulnerable 86-4.
Mark Wood, who has troubled all of Australia's batsmen with his pace this series, got the wickets of Warner (3) and Labuschagne (29), both caught behind by Ollie Pope, who had replaced injured wicketkeeper Jos Buttler.
Resuming today at 258-7, England could only muster another 36 runs as Nathan Lyon and Scott Boland quickly mopped up the lower order to establish a first-innings lead of 122.
Lyon (2-88) struck in the third over the morning, teasing Leach into a big shot but was easily caught by Pat Cummins for 10.
Boland, returning with the all-clear after needing a scan on his ribs following a fall while he was bowling just before tea on Friday, finally ended Jonny Bairstow's defiant innings at 113.
Bairstow faced 158 balls, hitting eight boundaries and three sixes. Boland then had Stuart Broad caught behind for 15 to end the innings.
Boland finished with 4-36 and now has 11 wickets in first three test innings.