Facing a target of 392, they slumped to 16 for two at stumps in Kingston, needing another 376 runs to win.
It seems only rain can save the home team from another heavy defeat after they lost the first test by nine wickets.
With more than two days left, Australia could have batted for another session at least in their second innings and still left themselves ample time to bowl out the West Indies.
But Clarke boldly declared with Australia at 212 for two wickets in their second innings, having earlier dismissed the hosts for 220 for a first innings lead of 179.
This gave Australia time to bowl eight overs before stumps, and they made a dream start when, in the third ball of the innings, Mitchell Starc bowled Kraigg Brathwaite for a duck.
Brathwaite chopped the ball onto his stumps, but it was a needless wicket to give away because the ball was sailing well wide of off stump and could have been left alone.
Worse followed for West Indies in the final ball of the first over when Starc had test debutant Rajendra Chandrika
caught at gully for another duck.
Chandrika, who slashed at a ball well wide of off stump, became just the second West Indies player to record a duck in both innings.
It could have been even worse for the hosts as Darren Bravo was given out lbw to off-spinner Nathan Lyon, but the decision was reversed when the replay showed the ball would have bounced over the stump.
Bravo (8) survived until stumps with Shane Dowrich (1).
Australia, who won the first test by nine wickets, are seeking to sweep the two-test series ahead of the Ashes against England, with the first test in Cardiff less than a month away.
Earlier in the day, all rounder Jason Holder, batting at number eight, was left stranded on 82 not out in the West Indies first innings.
Holder swatted the four Australian specialist bowlers to all corners of Sabina Park with an aggressive display of lusty hitting more reminiscent of a limited overs game than a test.
His unbeaten score, off only 63 balls, included 12 fours and two sixes and allowed West Indies to avoid the follow on.
Despite the daunting task, Holder has not given up West Indies can turn the game around.
"For me it’s still a good batting pitch," Holder told the host broadcaster. "You need to spend some time to get (settled) in. Once you get in it becomes a lot easier."
"It’s test cricket. You must expect pressure. We just have to apply ourselves for as long as we can tomorrow."
(Reporting by Andrew Both in Cary, North Carolina; editing by Martyn Herman, Toby Davis, Peter Rutherford)
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