Stuart Broad struck twice late in the day as Australia ended Day 4 at 107/3, still requiring a further 174 runs to win at Edgbaston on Monday.
Resuming at 28/2, the overnight pair of Ollie Pope and Joe Root added 49 more runs before the former was cleaned up with a magnificent yorker by Pat Cummins for 14.
Root featured in yet another crucial 52 run stand for the fourth wicket alongside Harry Brook before the former was stumped by wicket keeper Alex Carey off Nathan Lyon for a 55 ball-46.
Brook looked good in his innings of 46, before dragging a delivery from Lyon to Labuschagne who took a fine low catch at mid wicket.
Keeper-batter Jonny Bairstow scored 20 before being trapped leg before wicket by Lyon while attempting a reverse sweep.
Ben Stokes was then trapped right in front by his opposite number for 43. All-rounder Moeen Ali (19) fell to Josh Hazlewood, gloving a short ball to the keeper.
However some useful contributions by Robinson (27), Broad (10)* and James Anderson (12) lifted the hosts to 273, with an overall lead of 281.
In response, openers Usman Khawaja and David Warner got Australia off to a fantastic start. The duo added 61 runs for the opening wicket before the latter edged a delivery from Ollie Robinson to wicket keeper Bairstow for 36.
However Broad claimed two quick wickets to remove Marnus Labuschagne (13) and Steven Smith (6) as the Aussies ended Day 4 at 107/3, still requiring a further 174 runs to win on the final day.
Usman Khawaja was undefeated on 34 while nightwatchman Scott Boland was batting on 13.
Stumps at the end of Day 4
Australia 386 and 107 for 3 (Khawaja 34*, Boland 13*, Broad 2-28) need 174 more runs to beat England 393 for 8 declared and 273 (Cummins 4-63, Lyon 4-80)