Visiting captain Kraigg Brathwaite ended the day unbeaten on 101 to give his side, on 192-3 and trailing by 306 runs at the close, a fighting chance after Australia earlier declared their second innings closed on 182-2.
West Indies were bowled out for 283 in their first innings but Australia opted not to enforce the follow-on after smashing a massive 598 in their own first innings.
A fiery opening spell on Saturday by Alzarri Joseph, bowling with the sort of aggression that was missing on day one, had Marnus Labuschagne twisting and weaving early in the proceedings.
Caught at gully off a no-ball on 19, the zestful batsman capitalised on his good fortune and punished the weary fielders in an innings that became ever more belligerent as a lunchtime declaration loomed.
"I've certainly had the rub of the green on many occasions of late," Labuschagne told reporters.
"It must be all the prayers from my mum, my grandma and my family getting all set up at once."
Five balls before the break, Labuschagne became only the eighth batsman to register a double and single century in the same test, ending unbeaten on 104 from 110 deliveries.
With fast bowler Pat Cummins out of the attack due to quad soreness, the West Indies openers batted resolutely to guide their side to 84 without loss at tea.
Debutant Tagenarine Chanderpaul scored 45 before chopping Mitchell Starc on to his stumps, induced to drive through an open off-side field. Shamarh Brooks was next to fall, edging Nathan Lyon (2-54) to slip on 11.
Australia's bowlers laboured late into the afternoon as Brathwaite worked towards his 11th test ton, which he brought up with a sweep to deep midwicket, but Lyon struck late to send Jermaine Blackwood on his way for 24.
The home side head into the final day as firm favourites, but the Islanders will dare to dream that a draw can still be salvaged on a pitch without many demons.
"It's not going to be easy, but obviously I'm going to go out there and fight for the West Indies," Brathwaite said.
The highest successful test run chase is 418 by the West Indies against Australia at Antigua in 2003.
