Resuming on 378-6, Alex Carey made his way to 63 while the other overnight batter Michael Neser scored 16.
After dismissing both players, England were frustrated when Mitchell Starc and Scott Boland combined for partnership of 75 for the ninth wicket.
Starc was the more dominant of the duo and hit a valuable 77 while Boland just hung in for 21 not out, but soaked up 72 balls.
Brendan Doggett also contributed 13 as Australia went past the 500-mark to eventually be bowled out for 511, and take a lead of 177 on the first innings.
England now had the task of batting with the lights in full effect and lost Ben Duckett for 15 while Ollie Pope made 26.
Zak Crawley looked in decent form but the opener was caught by Neser off his own bowling to walk to teh dressing room for 44.
First innings centurion Joe Root could manage 15 before Starc had the England kingpin dismissed for 15 to leave the tourists on 121-4.
It was disastrous end to the day for the visitors when Harry Brook was caught behind off Boland.
England finished on 134-6, still trailing by 43 runs, with Ben Stokes and Will Jacks having a massive task on day four if they want to rescue the game.
England captain Harry Brook says his dropped off catch off Sanju Samson went a long way to his team losing their T20 World Cup semifinal against co-hosts India in Mumbai.
Glenn Maxwell says he will not make any short-term decisions about his future and is still hopeful of playing T20Is for Australia at the next World Cup in 2028.
Sam Curran feels England need to be on top of their game if they are to beat India and silence the home crowd in Thursday's T20 World Cup semifinal in Mumbai.
Will Jacks says England are relishing life under Harry Brook's captaincy after booking their place in the semi-finals of the 2026 T20 World Cup with a perfect Super Eights record.
