Nathan Lyon has landed a crucial late blow to set up a final-day assault for Australia in Perth, after the West Indies fought to stumps on day four at 3-192 in pursuit of an unlikely 498 for victory.
After giving themselves five sessions to bowl out the visitors, Australia were forced to work hard to claim only three wickets in two sessions on Saturday.
With Pat Cummins unable to bowl on day four as he nursed a quad injury, West Indies captain Kraigg Brathwaite offered the main resistance as he swept and drove his way to an unbeaten 101.
West Indies batsman Shamarh Brooks admitted after day three that any chance of a win might be "out of reach" for his team, as Australia built a big lead.
As it was, the target of 498 is 80 runs more than the record the West Indies set for the highest successful chase when they pulled off a miracle against Australia in 2003.
But at one stage on Saturday, West Indies would be forgiven for daring to dream again.
After an unbeaten Marnus Labuschagne century allowed Cummins to declare his side's second innings at 2-182 at lunch, the tourists worked their way to 0-116 in response after tea.
Mitchell Starc had beat the edge repeatedly in the opening session, while Josh Hazlewood was also unlucky .
Starc however was the man to finally get the breakthrough, routinely changing the field before Tagenarine Chanderpaul chopped on to his stumps for 45.
Brooks also came and went for 11 when he edged Lyon to slip, while Jermaine Blackwood survived one review and overturned an lbw call before he got off the mark.
But after Blackwood worked his way to 24 just before stumps and with the score at 2-191, Lyon had him edging to Labuschagne in close and the Queenslander took a smart catch.
That wicket could prove vital, with Australia now needing just seven more to retain the Frank Worrell Trophy and start the Test summer with a win.
Australia could also have Cummins back bowling by the second hour on Sunday, pending further assessment before play after he returned to the field in the final session.
Earlier, Labuschagne became only the eighth player in Test history to score a double century and another ton in the same match.
Streaky again, Labuschagne at one stage top-edged Alzarri Joseph for six and was then caught off his bat and helmet from the next delivery from the quick before a no-ball was called.
But he went from 50 to 104no in 40 balls before the declaration, including two straight drives off Jayden Seales and a six back over Roston Chase's head.
It came as the wheels fell off the Windies' attack, with Kyle Mayers (shoulder) and Kemar Roach (thigh) both unable to bowl.
Meanwhile, David Warner cut a frustrated figure when he was caught at bad pad to Chase after reaching 48 with apparent ease.
After admitting earlier in the week he needed runs to keep the pressure off, the 36-year-old flung his head back and trudged off on his dismissal.