Tim Southee (four for 52) was the star man on the third morning in Dunedin as the Black Caps dismissed the tourists for 213 in response to their first-innings 609 for nine declared, with only Shivnarine Chanderpaul (76) standing up to the challenge.
Trent Boult then struck to remove Kieran Powell as the West Indies followed on, but Bravo (72no) and Kirk Edwards (59) produced a stand of 117 to help the tourists close on 168 for two, still 228 shy of making their hosts bat again.
Marlon Samuels survived via DRS in the third over of the day after initially being given out lbw to Boult but three balls later the hosts were legitimately celebrating their first wicket of the morning.
Bravo drove at Southee and was caught by Brendon McCullum at gully for 40 with the tourists having added just three runs to their overnight 67.
Samuels followed three overs later, caught at slip by Ross Taylor off Southee, and the same pair combined for the third wicket of the morning in the first over after drinks.
This time Taylor needed two bites of the cherry, but pouched it safely nevertheless, to remove Narsingh Deonarine and reduce the West Indies to 106 for five.
It was left to Chanderpaul to mount a rescue operation.
He struck a boundary to take him past 11,000 runs in Test cricket, then plundered 14 off one Ish Sodhi over to reach his half-century shortly before lunch.
Chanderpaul drove streakily for four off Boult after the resumption, but later in the same over the bowler had more luck when Denesh Ramdin gloved a catch behind while trying to leave the ball, ending a partnership of 68.
In his next over, Boult trapped Chanderpaul lbw not offering a stroke and, despite a flurry of strokes from captain Darren Sammy, the final three batsmen could offer little to boost the West Indies' total as Sodhi picked up a couple of cheap wickets.
And there was more of the same second time around, with Powell's wicket - edging Boult to Southee at third slip - the key moment in the 14 overs that were possible before tea.
It was a different shorty in the evening with the batsmen beginning to look comfortable, Edwards lifting Sodhi for six and Bravo claiming three successive boundaries off Southee.
Bravo survived an appeal for caught behind with the umpires unable to be certain even with the use of DRS that the ball had touched his glove.
He drove his way past 50 and Edwards followed him after drinks, but a Sodhi googly did for the latter before stumps.