An 84th-minute penalty from Perth's Steven McGarry had looked set to see the game end level after his strike cancelled out a maiden A-League goal from Central Coast defender Trent Sainsbury.
Glory midfielder Jacob Burns was then sent off in injury time after a series of incidents culminated in the midfielder being shown two yellow cards.
But McGlinchey stole the headlines in the 96th minute, when he was in the right place at the right time to grab a win that lifts the Mariners to fourth in the table.
The New Zealand midfielder blasted the ball home from close range after veteran striker Daniel McBreen’s header had hit the post and fallen kindly for him.
Perth's lack of early possession meant that the home side were in control for most of the fixture, and prior to the winner the Mariners spurned a string of chances to make life more comfortable for themselves.
However, injuries to key defenders Sainsbury and Zachary Anderson may still leave home coach Phil Moss concerned despite the victory.
‘Even though it was late in the game, the Arsenal fans started singing my name. It felt like the moment I announced myself as a professional footballer’: Ex-Gunners star recalls scoring first senior goal as a teenager at Highbury
'I started to drive myself crazy': Former Manchester United midfielder explains why it all went wrong for him under Erik ten Hag
