Taylor, 33, whose first century was also at Seddon Park against England in 2008, said before the Test that Crowe wanted him to break his record.
New Zealand declared their second innings at 291 for eight, with Taylor 107 not out, and then took two top-order wickets by stumps.
It took the 33-year-old Taylor 83 Tests to get to his 17th century compared to 77 for Martin Crowe and 61 for Kane Williamson.
The new ball pair then took four wickets between them on a benign pitch as the West Indies plummeted to 215 for eight at stumps.
Southee featured in a 61-run stand for the 10th wicket with Trent Boult and later removed Kieran Powell without scoring in his first over.
New Zealand punished some wayward bowling to reach154-1 before an initial slump saw them fall to 189-5.
Gabriel and Kemar Roach struggled to find a consistent line and length when the ball was new and lacked the hostility to trouble the Kiwis.