Australia's hopes of pushing for a series-clinching victory were thwarted by a torrential storm at Queen's Park Oval in Trinidad.
Most of the day was lost, leaving Australia 127 runs ahead with seven second innings wickets in hand.
With more showers forecast for Thursday, a draw was looming as the most likely result, which would be enough for Australia to retain the Frank Worrell Trophy after the tourists won last week's first Test in Barbados.
But Australia opening batsman Ed Cowan said his team were not giving up hope of winning and may even be prepared to roll the dice and risk defeat to give themselves a chance of victory.
"I don't think there's any doubt about that," Cowan said.
"We'll open up the game at some point, probably willing to lose it for the chance to win it. It will depend on whether they're willing to risk the game, I guess."
If the weather stays fine, Australia's best chances of forcing a result would be to try and add another 100 runs before lunch then set up an early declaration that West Indies would be tempted to chase.
"I'm not the captain. I haven't spoken to (coach) Mickey (Arthur) or (captain) Michael (Clarke) but on that wicket we've seen scoring rates at a little over two for the game so 230, 240 runs even off 65, 70 overs (would be tough)," Cowan said.
"That's long enough for us to bowl them out if they're willing to have a go. We'll see. I don't quite know how the brains' trust is playing it. That's what I'd be thinking."
West Indies fast bowler Kemar Roach, who captured all three Australian second innings wickets to take his total to eight for the match, said the home team would also press for a victory but the prospects were dim.
"It's kind of dicey now but we'll see how it goes," he said.
"We'll devise a plan tonight for tomorrow and come out to look to compete as well.
"We look to win every game, that's our mentality. We're going to come out tomorrow and whatever we do, we're going to do it in a positive way."
Roach has been the standout bowler for West Indies and once again provided the breakthroughs when his team needed them.
He dismissed David Warner and Shane Watson [ Images ] in his first over, knocking Watson's off-stump out of the ground.
Roach removed Cowan for 20 before the rain washed out the rest of the day, just when the Australian was starting to look dangerous.
Australia 1st innings 311
West Indies 1st innings (overnight 252-9)
A.Barath lbw b Beer 7
K.Brathwaite lbw b Hilfenhaus 0
K.Powell lbw b Pattinson 19
D.Bravo lbw b Hussey 38
S.Chanderpaul lbw b Lyon 94
N.Deonarine st Wade b Lyon 55
C.Baugh lbw b Beer 21
D.Sammy c Hussey b Lyon 1
S.Shillingford c Cowan b Lyon 4
K.Roach c Wade b Lyon 0
F.Edwards not out 0
Extras (b-1 lb-8 w-1 nb-8) 18
Total (all out, 104.4 overs) 257
Fall of wickets: 1-0 2-26 3-38 4-100 5-230 6-231 7-237 8-241 9-249
Bowling: Beer 25.4-9-56-2, Hilfenhaus 16-4-39-1, Lyon 29-9-68-5, Pattinson 11-2-40-1 (nb-4), Hussey 6-1-19-1 (w-1), Watson 12-5-14-0, Warner 3-1-9-0, Clarke 2-0-3-0
Australia 2nd innings
E.Cowan lbw b Roach 20
D.Warner c Bravo b Roach 17
S.Watson b Roach 0
R.Ponting not out 32
M.Clarke not out 3
Extras (nb-1) 1
Total (for three wickets, 30 overs) 73
Fall of wickets: 1-26 2-26 3-57
Bowling: Edwards 5-2-13-0 (nb-1), Shillingford 15-3-33-0, Roach 10-1-27-3