The Glory's new Argentine import Matias Cordoba put the visitors in front and teen striker Adam Taggart pulled one back for the Jets before Thwaite sealed victory in the 92nd minute.
The Jets remain in fifth place but have a congested bottom-half of the ladder breathing down their neck. Glory meanwhile climb to seventh.
Having already played their round-26 match against Wellington due to New Zealand's international commitment, time is fast running out for the Jets who only have two games remaining before the finals.
With a game away to fourth-placed Adelaide next weekend before facing rampant leaders Western Sydney, the encounter against the Glory was their best shot at picking up three points.
Desperate to make amends for their 5-0 thrashing by Melbourne Victory last weekend, the Jets came out firing with young gun Taggart, playing at the point of attack in place of injured marquee man Emile Heskey, forcing Perth 'keeper Daniel Vukovic into action just four minutes in.
But it was the Glory's newest recruit Cordoba who broke the deadlock in the 24th minute when he managed to get on the end of a Chris Harold cross for his first A-League goal.
Socceroo tyro Taggart levelled just before the break, slotting home off a Marko Jesic through-ball.
Glory skipper Jacob Burns made his long-awaited return after a six-week injury layoff as a second-half substitute for Cordoba.
Swiss defender Dominik Ritter went within a whisker of putting the Jets in front in the 62nd minute, Vukovic doing well to save the effort.
The Glory gloveman produced yet another great save, stopping a strike from English attacker Michael Bridges with less than 20 minutes remaining.
Newcastle's Mark Birighitti made one of his few saves of the game moments later, defusing a dangerous header from the A-League's all-time leading scorer Shane Smeltz.
But it was Thwaite who sealed the win, pouncing on a goal-mouth scramble late in injury time to get his first goal of the season.