The Taipans fifth straight win lifts Cairns to third while the Breakers are sixth, outside the top four playoff spots.
The Snakes' have a 12-9 record and are now in line for their first playoff action since the 2016/17 season.
The Breakers led 78-71 with half a quarter to play, but Cairns fought back to close out the game on a 14-1 scoring run.
Taipans coach Mike Kelly said this scalp was massive in terms of their playoff push.
"That was a massive game tonight. New Zealand is playing good basketball: they run good stuff, they attack, they're really tough to defend, their bigs can shoot and they've got guys who can get on the rim," he said.
"That would be way ahead of ourselves to start saying we're in the finals, but we can see it from here. And the belief is building."
The Breakers' playoff hopes are now hanging by a thread, with Sunday's home game against Brisbane a must-win.