West Indies need a further 243 runs with all 10 wickets in hand to win the opening Test against New Zealand in Auckland.
The tourists ended the third day at Eden Park on Saturday on 48 without loss, chasing 291 for their first Test win in over nine months.
Chris Gayle was unbeaten on 28 with fellow opener Daren Ganga not out 20 and the temporary drop-in pitch, which had been notoriously difficult for the first two days, suddenly showing signs of flattening out.
"If someone told us at the start of the Test match that we had to score 240-odd with two days left, we would happily grab it," Ganga told the New Zealand Press Association.
"The guys have worked hard and we're really pleased with the position we're in."
New Zealand were bowled out for 272 midway through the final session after starting the day on 98-4 with an overall lead of 116.
Wicketkeeper Brendon McCullum top-scored with 74 off 126 balls while Stephen Fleming and Daniel Vettori both made 33.
History is still with the home team. It's been nearly 40 years since a side has scored more than 291 in the fourth innings to win at Eden Park and Ganga said the visitors weren't taking anything for granted.
"We don't want to talk too quick. We know it's going to take a lot of hard work," he said.
"I'm sure the guys are up to the challenge. They're very eager to make this one count."
New Zealand were in all sorts of trouble when they lost their skipper Fleming (33), nightwatchman James Franklin (20) and injured batsman Nathan Astle (13) before lunch to slump to 146-7 when McCullum and Vettori came to the rescue.
McCullum struck seven boundaries, and a pulled six off Fidel Edwards that sailed over the rope, to register his sixth Test fifty.
He also shared a 64-run partnership with spinner Vettori and a 62-stand with Shane Bond, before Gayle (4-71) polished off the tail with his under-rated spin then padded up with Ganga to give the West Indies the perfect platform to push for victory.
Scorecard
New Zealand (1st innings): 275 (S. Styris 103no, N. Astle 51)
West Indies (1st innings): 257 (R. Sarwan 62, D. Bravo 59)
New Zealand (2nd innings): (Overnight: 98-4)
H. Marshall c Ganga b Bradshaw 1
J. How c Ramdin b Bradshaw 37
P. Fulton b Edwards 28
S. Fleming lbw b Bradshaw 33
S. Styris c Bradshaw b Edwards 5
J. Franklin b Gayle 20
N. Astle run out 13
B. McCullum c Bravo b Gayle 74
D. Vettori c sub b Gayle 33
S. Bond not out 18
C. Martin b Gayle 0.
Extras: (lb-3, nb-5, w-2) 10.
Total: (all out, 103.1 overs) 272
Fall of wickets: 1-11, 2-66, 3-73, 4-88, 5-118, 6-143, 7-146, 8-210, 9-272, 10-272.
Bowling: F. Edwards 21 - 3 - 65 - 2 (nb-4), I. Bradshaw 34 - 10 - 83 - 3 (nb-1 w-1), J. Taylor 1 - 0 - 6 - 0, C. Gayle 30.1 - 5 - 71 - 4 (w-1), D. Smith 17 - 6 - 44 - 0.
West Indies (2nd innings): (Target: 291 runs)
C. Gayle not out 28
D. Ganga not out 20.
Extras: 0
Total: (for no loss, 18 overs) 48
To bat: R. Sarwan, B. Lara, S. Chanderpaul, D. Bravo, D. Smith, D. Ramdin, I. Bradshaw, F. Edwards, J. Taylor, R. Morton.
Bowling: S. Bond 4 - 0 - 20 - 0, J. Franklin 4 - 0 - 10 - 0, C. Martin 4 - 2 - 13 - 0, D. Vettori 5 - 4 - 5 - 0, N. Astle 1 - 1 - 0 - 0.