In a bizarre finish to the match, the Wildcats hit the lead with a minute remaining after Breakers forward Gary Wilkinson accidentally tipped the ball into his own basket.
Then, with the clock counting down, Breakers guard CJ Bruton charged in for what he hoped would be the game-winning lay-up, only for Redhage to come over the top with a massive rejection to secure the win.
Redhage had endured a dirty night up until that point, but the forward erupted in pure joy after pulling off the game-deciding move in front of a sellout crowd.
Perth's win levelled the series at 1-1, with the decider to be played in Auckland next Tuesday.
Wilkinson top-scored for the Breakers with 29 points, while CJ Bruton (20 points) and CJ Pledger (14) were also important for the visitors.
The Wildcats hit just one of 18 three-point attempts, but still managed to come away with the win courtesy of an even scoring load.
Star guard Kevin Lisch and Jesse Wagstaff each scored 15, while Matthew Knight (13), Greg Hire (12) and Redhage (12) also chipped in.
Wilkinson was unstoppable in the first half, with the centre nailing 18 points on 8-of-10 shooting to silence the Perth crowd early.
Bruton was deadly from long range as the Breakers opened up a 46-33 lead midway through the second quarter.
Tempers threatened to boil over late in the half as the teams engaged in a push-and-shove, and the extra heat seemed to help the Cats as they closed the deficit to six by the main break.
However, the half ended on a controversial note when Wilkinson was awarded a basket on the buzzer despite Wildcats players believing time had already elapsed.
The Wildcats failed to hit a three-pointer in the opening half, and it took them until the dying minutes of the third term to finally register one.
Several bombs from Bruton in the final term looked to be enough to give NZ the win, but the Wildcats kept plugging away and eventually wore down the Breakers.
Star NZ swingman Tom Abercrombie managed just six points in his first game back from an ankle injury.
"I thought he was going to make the shot," Wildcats coach Rob Beveridge said of Bruton's final play.
"But Shawn came out of nowhere. He was unbelievable. He came out of nowhere and got it. We were lucky there."
Bruton said he thought he saw a gap where he could sneak in for the shot.
"When you've got Matty (Knight) on you, he's a big fellow, so I thought I'd drive around him," Bruton explained.
"When I did that I thought: 'oh yeah'. And when I took the shot - I told the coach I should have taken the sky hook.
"But I didn't and he (Redhage) got a piece of the ball, and the end result is they won and we move on."