The Windies finished the day at 6-443, forging a lead of 47 over the hosts after starting the day 228 runs behind.
The 24-year-old Bravo, a cousin of the great Brian Lara, batted for more than nine hours in his unbeaten 210 to defy New Zealand's bowlers on the lifeless University Oval pitch.
He punched the air in delight after reaching 200 and was embraced by captain Darren Sammy as he became the 26th West Indian to pass the milestone.
"It's a great feeling. I set myself to bat out the day. I backed myself and the team was backing me as well.
"I just went out and enjoyed my game in the best possible way," he said.
But New Zealand leg-spinner Ish Sodhi said they had not lost heart despite claiming only four wickets for the day and dropping three catches in the process.
"On the scoreboard, we still are on top and Test cricket's not an easy game.
"We created enough opportunities. They were tough ones at times, and that's what we've got to judge ourselves on," the 21-year-old said.
Bravo found useful allies in Narsingh Deonarine (52), Denesh Ramdin (24) and Sammy (44 not out) to frustrate the home side seeking their first Test win of the year.
Starting the day on 2-168 after following on - having been dismissed for 213 in reply to New Zealand's 9(dec)-609 declared - the tourists lost two early wickets to dent their hopes of saving the Test.
But Bravo, who began the day on 72 after being given not out by the television umpire on the third day after appearing to be caught behind, dug in and mixed gritty defence with powerful drives through the off-side.
The Blacks Caps might have expected to make short work of the opposition when Tim Southee struck in the fourth over to remove Marlon Samuels.
Those hopes were further boosted when Neil Wagner took his first wicket of the Test to claim the prized scalp of veteran Shivnarine Chanderpaul.
The 39-year-old, who a day earlier became just the seventh batsmen to pass 11,000 runs in Tests, was trapped lbw for one to reduce the tourists to 4-185.
But Bravo and Deonarine combined for a 122-run stand for the fifth wicket before Corey Anderson's first Test wicket on home soil removed Deonarine.
Bravo is averaging 59.64 away from home, the best record by a West Indian.
He added 56 with Ramdin and combined with Sammy in a unbroken stand of 80.