West Indies, who closed on 135 for four, will be hoping to wrap up the win at Sabina Park early on the fourth day, allowing them to join the rest of Jamaica in watching Usain Bolt and his team-mates in the Olympics 100 metres final.
In order to enjoy the televised action from London in positive mood, the Caribbean side will probably need the experienced Shivnarine Chanderpaul to see them through to victory on Sunday.
Unusually in the three innings so far in this short series, Chanderpaul has not made a major contribution but nothing will sooth any nervous jitters from his team like the presence of the Guyanese left-hander.
Chanderpaul ended the third day on 20 not out, partnered by nightwatchman Kemar Roach after the late dismissal of Marlon Samuels - a wicket that will give the Kiwis some hope of pulling off a surprise series-tying win.
Samuels, who hit a century in the first innings, was again looking good on 52 when he was superbly caught by New Zealand skipper Ross Taylor in the slips off Doug Bracewell.
The wicket restored some confidence to the tourists after they had gained early breakthroughs with the removal of openers Chris Gayle (8), trapped lbw by Trent Boult, and Kieran Powell (6) in similar fashion to Tim Southee.
The West Indies innings was halted briefly to allow spectators and players to watch the women's 100 metres final from London which, to the delight of the crowd, was won by Jamaican Shelly-Ann Fraser-Pryce.
Earlier, the tourists had resumed on 59 for two, with a 110 run advantage, but not for the first time on what has been a disappointing tour of the Caribbean, New Zealand's batting let them down.
Deonarine, considered very much the back-up spinner to Sunil Narine, added two more wickets to his overnight pair - removing Brendon McCullum (19) and Kane Williamson (8).
Narine was not to be overshadowed however, chipping in with three for 19 as he got through the lower order cheaply while paceman Tino Best started the morning by removing nightwatchman Neil Wagner and then the most crucial breakthrough of the session - removing skipper Taylor for a duck.