Bravo was on 72 while Marlon Samuels was on 17 after Edwards was trapped in front by legspinner Ish Sodhi for 59 about 45 minutes before the end of play as the West Indies adopted a more resolute approach in their second innings.
The visitors were 168 for two at the close but still facing a big defeat after New Zealand captain Brendon McCullum had enforced the follow on before tea when they dismissed West Indies for 213 in their first innings.
New Zealand, who made 609-9 declared in their innings, have not won a Test in more than a year, having drawn five and lost four of their matches since McCullum took over the captaincy in late 2012.
Trent Boult had plunged the visitors into the mire before tea when he captured the wicket of Kieran Powell, well caught by pace bowler Tim Southee at third slip for 14, before Bravo and Edwards combined for a 117-run partnership.
"We always knew they would put up a fight and that's what they have done," Boult told Radio Sport pitchside about the visitors' better performance in their second innings. "We can't get frustrated.
"We understand they will put together some partnerships but its all about us executing our plans keeping that pressure on."
The day's play had begun under a cloud with the news that the International Cricket Council was investigating three former New Zealand players for match-fixing, although the current crop of players failed to allow that to distract them.
Their bowlers kept pressure on the West Indies batsmen all day with Southee taking the wickets of Bravo (40), Marlon Samuels (14) and Narsingh Deonarine (15) before lunch.
The visitors resumed after lunch on 159-5 and then Boult produced a fiery spell to capture wicketkeeper Denesh Ramdin for 12 and then Shivnarine Chanderpaul for 76 in his next over.
Chanderpaul, who passed 11,000 career runs before lunch when he belted Neil Wagner to the boundary to reach 37, was the key to the first innings and when he shouldered arms to a Boult delivery and was trapped in front, it was only a matter of time before they were bowled out.
Captain Darren Sammy, however, produced a rearguard 27 not out despite being severely hampered by a hamstring injury he sustained while bowling.
"Nice team effort. I think the pressure was maintained by all ends and as a bowling group that's what we're trying to do," Boult added.
"It would have been nice to have one or more two (down) but that's the nature of the game.
"It is a little bit frustrating and there's still a long way to go in this match with a lot of work to do tomorrow and the next day."