New Zealand were all out at tea for 349, 18 runs short of the West Indies first innings' 367.
The prolific Taylor, who has taunted the West Indies bowlers in all three Tests, was dismissed for 131, while Narine in his first Test of the series returned figures of six for 91.
It was only the second time Taylor has been removed in four innings in the series in which he has amassed 493 runs.
Taylor's 11th Test century also saw him become only the second New Zealand player to score centuries in three consecutive Tests after Mark Burgess achieved the same feat between 1969 and 1972.
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Taylor was at the crease for 326 minutes in an innings that included 16 fours and two sixes, with the bulk of his runs the result of deftly working the ball into gaps for ones and twos.
McCullum was in for just over seven overs in the morning and scoring just one run before showing his frustration by slashing at a sharply turning offbreak from Narine to be caught in slips by Darren Sammy.
All-rounder Anderson, playing only his fifth Test, launched into dangerman Narine and took 13 off one over on his way to 39 before he was undone by spin from the other end.
BJ Watling chimed in with 20 before he was caught behind off Sammy, leaving the impressive Narine to remove Tim Southee (18), Ish Sodhi (nine) and Neil Wagner (22) to wrap up the innings.