The game was delayed half an hour due to traffic issues in Perth and the Breakers arrived at the arena 55 minutes later than scheduled.
After a dour opening, halftime greeted the sides as the Breakers held a 35-32 lead but both teams, who could meet in the playoffs in two weeks, shot less than 35 per cent from the floor in the first half.
The Wildcats, spurred on by a home crowd of 12,000, started the third period with gusto as they chopped the Breakers' lead down.
Breakers centre Ekene Ibekwe, who eventually fouled out during the first period of overtime, finished with 11 points, seven rebounds and six blocks.
The scores were level at 62 as the game entered the final four minutes. Both sides were struggling with their shooting as the playoff-style defence was on display at both ends of the floor.
With 10 seconds left in regulation time, the scores were locked at 68 each but Jackson's potential game-winner didn't fall and extra minutes were needed.
The first overtime went back and forth and Beal hit a buzzer-beat to send it to another extra period at 80-all. Beal's freebies nearly did it in the second period of OT before Jackson's miracle stole the show.
DeAndre Daniels had 24 points for Perth, while Corey Webster had a career-high 27 for the Breakers.
The Breakers will finish the regular season with a home outing against Cairns at Vector Arena on Sunday.
New Zealand Breakers 89 (Corey Webster 27) Perth Wildcats 87 (Jermaine Beal 24, DeAndre Daniels 24) 1Q: 18-16, halftime: 35-32, 3Q: 48-55, 4Q: 68-68, 1OT: 80-80