Glory appeared to have secured a point through a 84th minute penalty to Steve McGarry in his 100th appearance for Perth.
But in a frantic five-minute stoppage time period, Perth had skipper Jacob Burns sent off for a second yellow card offence.
With just seconds remaining, New Zealander McGlinchey, in his second last game for the Mariners before starting a loan spell in Japan, fired home from close range after Daniel McBreen had hit the post.
The victory lifted Central Coast two spots to fourth and left Perth in eighth.
It was an eventful game for the Mariners' central defenders, with Zac Anderson and Trent Sainsbury combining for the latter's first A-League goal in the 12th minute, before both were forced off with injury in the second half.
Substitute centre back Marcel Seip was booked for coming onto the field before the officials were officially notified and then gave away the penalty with a foul on Chris Harold.
Prior to stoppage time, the Mariners had a great chance to win the game in the 90th minute, but out of sorts striker Daniel McBreen fired wide.
The Mariners had 64 per cent of possession in the first half and were good value for their 1-0 lead at the break.
Sainsbury, whose only previous goal for the Mariners was in an Asian Champions League game, opened his A-League account in his 56th domestic appearance.
He fired home following a flick from central defensive colleague Zac Anderson, after Glory only partially cleared a 12th minute corner.
The Mariners created several other chances - Josh Rose and McBreen shot straight at Glory goalkeeper Danny Vukovic, while an unmarked McBreen stabbed the ball wide on another occasion.
Glory made the occasional incursion into the Mariners' penalty area, with Brazilian Sidnei firing wide in the 15th minute and a couple of other dangerous crosses not being rewarded.
Glory came into the game more in the second half and the Mariners struggled to find a telling final ball.