Having resumed on 91-6, the tourists rallied with Darren Bravo (81) and debutant Carlos Brathwaite (59) adding 90 but the Windies were 271 all out.
Australia, with a first-innings lead of 280, did not enforce the follow-on.
Instead, Steve Smith and Usman Khawaja both hit half-centuries as Australia closed on 179-3, a lead of 459 runs.
Brathwaite provided rare moments of cheer for the beleaguered West Indies, who were beaten by an innings and 212 runs inside three days in the first Test.
The Barbadian made the most of two strokes of good fortune before hitting Nathan Lyon for 11 in one over to reach his half-century.
The 27-year-old was twice given out to James Pattinson, once bowled and once caught at fine leg, but was recalled on both occasions because television replays showed the Australian had bowled no balls.
Brathwaite was finally out when he spooned a return catch to Lyon (4-66), while Pattinson had Bravo caught by Smith at gully to finish with 4-72.
Kraigg Brathwaite caught Joe Burns at second slip off skipper Jason Holder before the unrelated Carlos had David Warner caught at gully to leave Australia 46-2 in their second innings.
However, captain Smith reached the close unbeaten on 70, after putting on 77 for the third wicket with Khawaja (56), who was out caught behind off Holder.
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West Indies avoided humiliation on day three of the second Test against Australia but are still facing a heavy defeat in Melbourne.