With rain threatening to eat into the final day's play the Black Caps took the majority of the morning session to take the four wickets they required to wrap up the tourists' second innings for 507.
While Bravo was crucially one of those early on, adding just eight to his overnight 210 before Trent Boult got a ball to keep slightly low, Darren Sammy (80) took over the fight to build the Windies' lead into a tricky 111 runs.
With conditions clouding over at the University Oval the home side's need to begin the task of chasing the runs became even more urgent, but the 10-minute mini-session before lunch went against them with Peter Fulton falling in the final over.
West Indies began the day 47 runs ahead after being asked to follow on 386 runs in arrears midway through day three.
Bravo's maiden Test double-century meant they at least went into the final day with some hope of pulling off a memorable draw.
That bid was therefore dealt a significant blow when Boult snuck a delivery through Bravo's defences in just the third over of the day.
Sammy took up the fight though in a counter-punching innings that had him in sight of just a second Test century.
The all-rounder added 38 runs for the eighth wicket with Shane Shillingford, and at one point launched leg-spinner Ish Sodhi for six, before his partner edged Neil Wagner (three for 112) to slip.
Sammy had reached 80 with two wickets still in hand, and in sight of lunch as well as three figures, only for the last two wickets to fall with the score on 507.
Wagner squared up Tino Best to induce another edge that Ross Taylor pouched.
With New Zealand taking the third new ball Sammy tried to take the long handle but only served to spoon a catch off Tim Southee to end the innings.
Shillingford then had Fulton caught behind, a decision given via DRS, to leave New Zealand three for one at lunch.