The tourists will need to survive the entirety of day five in order to end Australia's pursuit of an Ashes whitewash in Sydney.
Khawaja, who bagged 137 runs in the first innings, delivered a dashing total of 101 not out at his old stomping ground.
When Australia declared on 265/6, they left England staring at a near-insurmountable target of 388 - exactly 100 more than the record chase at this ground, and 91 more than their best total so far in the series.
One statistic which sums up England's incompetence is the fact that Khawaja - who is playing in his first Test of this series - has already score more runs than any other English player, barring the captain Joe Root.
The opening pairing has been the biggest problem of all. But Haseeb Hameed and Zak Crawley, both fearfully short of runs, played out 11 overs to reach 30 without loss. Remarkably, that represented England's best first-wicket stand of the trip.
Australia should get another 98 overs on Sunday, with England carrying three injured men in their top seven. Ben Stokes has had scans on a worrying side complaint, while Jos Buttler (left index finger) and Jonny Bairstow (right thumb) have both had X-rays.
In the absence of the latter pair, Ollie Pope was drafted in as substitute wicketkeeper and acquitted himself impressively with four catches, but everyone will be needed to play their part with the bat.
England began the day with centurion Bairstow still at the crease and three wickets up their sleeve as they looked to chip away at an overnight deficit of 158.
They were only able to take another 36 off, with Scott Boland removing Bairstow for 113 and adding Stuart Broad to continue his remarkable introduction to the Test arena.
Root tossed the ball to Wood while it was still fresh, just five overs in, and he responded by immediately snagging David Warner's outside edge. Pope took it safely and the job was under way.
Pope's second catch was more impressive, standing up to Jack Leach's spin, staying low and snapping up a sharp edge from Marcus Harris. By the time they added the prize scalps of Marnus Labuschagne and Steve Smith Australia were creaking at 86 for four.
For a moment, there was a hint that Australia could be toppled but Khawaja went on to lead a superb stand of 179 with Cameron Green.
A declaration looked certain but Pat Cummins gave Green the chance to sprint for his own ton. He got as far as 74 before slogging Leach straight in the air. The spinner made it two in two when Pope's smart reactions condemned Alex Carey to a golden duck, but Cummins refused to allow the hat-trick opportunity - waving Khawaja in from the balcony.
Australia have made a habit of new-ball wickets but found Crawley (22no) in breezy form, while Hameed hung in for eight.