Cotton, the NBL MVP favourite, overcame a scoreless opening term on Tuesday - on the back of a poor game against Adelaide last Saturday - to smoke 26 of his game-high 31 points after halftime.
His brilliant performance featured a freakish off-balance, double-pump three-pointer to force overtime, which he dominated.
"We had a timeout left and the other guys knew the play and executed the play great [with 1.7 seconds left in the final quarter]," Perth coach Trevor Gleeson said.
"BC with the double-pump, it looked like it might have been going a bit short, but we'll take it."
The Wildcats looked miles off the pace early, missing 20 of their first 23 shots, Cotton scoreless until midway through the second stanza and fellow import John Mooney (three points at 8%) enduring a rare stinker.
The Breakers, paced by in-form forward Finn Delany (25 points, nine rebounds), had applied the early heat and led 23-9 at quarter-time.
Then Perth flipped the script in the second, outscoring NZ 23-10 to reduce the deficit to one point at halftime before pinching the lead early in the third on Mitch Norton's corner three.
In the NBL's first fixture in Tasmania since 2005 and first in Launceston since 1982, the Wildcats led 69-63 with a minute left in regulation on a Todd Blanchfield steal and lay-up from a third-gamer Will McDowell-White's turnover.
But just when the game seemed over, McDowell-White made amends with a strong drive and a pair of free throws with 1.7 seconds remaining.
He deliberately tried to miss the second but it banked in, which gave Perth time to set up and erase New Zealand's three-point advantage.
Cotton did so in spectacular fashion on the full-time bell - after he had uncharacteristically missed his previous 17 three-pointers.
Cotton, who scored 20 of Perth's last 26 points, single-handedly outscored New Zealand 9-7 in overtime, further cementing himself as one of the NBL's all-time greats.
"This is a reminder that games can be decided by very small things," Breakers coach Dan Shamir said.
"We wanted to miss that last free throw, which is not always easy.
"Cotton then made a crazy three and here we are with the loss."
