Glory entered the game having not won any of their last four, and they could have suffered an early deficit when Bruno Fornaroli latched onto a mistake from Aryn Williams, but the forward's shot went wide.
The opening 20 minutes of the fixture were frantic as both teams had decent chances to break the deadlock, but it was City who made the opportunities count when Erik Paartalu nudged the ball inside the near post after receiving the end of Aaron Mooy's cross.
Both sides seemed to adopt the counter-attack approach, and it paid off for Glory as they won a free kick in a dangerous area two minutes into injury time.
After the break, City took the lead again when a decent one-two between Mooy and Fornaroli resulted in the former sliding the ball home thanks to a slight deflection off Ante Covic.
Twelve minutes alter, though, Chris Harold took advantage of Sorensen being out of position by unleashing a shot after latching onto a loose ball.
After coming from behind twice, Perth were dealt a blow when Collard was sent off after receiving a second yellow card for kicking the ball away, but the hosts managed to see out the game and take a point.
City sit fourth in the ladder, while Glory are eighth.