The 29-year-old right-hander has scored 493 runs in four innings and is within sight of Andrew Jones's New Zealand record of 513 runs in a three-match series. Taylor's series average is 246.5.
Narine has taken 18 of his 21 test wickets against New Zealand, and his bowling of Ish Sodhi for nine completed his second five-wicket haul in tests.
He had earlier trapped Tim Southee lbw for 18 after Darren Sammy had broken Taylor and BJ Watling's partnership when the wicketkeeper was caught chasing a wide delivery and edged to counterpart Denesh Ramdin.
Neil Wagner took the attack to the West Indies bowlers to ensure the hosts got close, but he was Narine's sixth victim when he skied a catch to Kirk Edwards for 22.
New Zealand had been in danger of conceding a large first innings lead to the visitors, who had scored 367, when Brendon McCullum was caught by Sammy at slip off Narine to reduce the hosts to 174 for four.
Corey Anderson joined Taylor and while he initially scratched around against Narine, he soon grew more comfortable against the 25-year-old Trinidadian, smashing him to the fence three times in one over.
Anderson, however, fell in the next over by Veerasammy Permaul when Narsingh Deonarine took a well-judged catch at deep midwicket after the all-rounder and Taylor had combined for a quick 50-run partnership.
(Reporting by Greg Stutchbury in Wellington; Editing by Ian Ransom)
