Perth went into the game in seventh place with 13 points from their first 10 fixtures and would have looked at this match as an easy victory.
But AIS had other ideas as they took points from an opponent for just the second time this season following their draw against Sydney in Round 9.
They claimed the win courtesy of two late second-half free kicks from Matthew Schmidt and Joshua Barresi.
First, Schmidt scored a screamer over the wall and into the opposite corner to level the scores before a Barresi free kick bounced just in front of Glory keeper Jordan Frankell and slipped through his fingers into the back of the net.
Perth had taken the lead in the 70th minute through Andrew Higgins when he produced a smart finish.
The Glory then squandered a number of chances to double their advantage before the visitors scored their two goals.
Babor Sam went close with an effort that cannoned off the post, while Blamo Quaqua also narrowly missed.
In a scoreless first half it was Perth who had the better of the chances as they used the width of the pitch to attack their opponents.
But the AIS were not without chances of their own and had an appeal for a penalty turned down after a dangerous tackle in the box.
First-team Glory substitute Chris Harold forced a good save, before a spectacular bicycle kick from Jacob Collard was also kept out from the resulting corner.
Perth 1 (Higgins 70)
Australian Institute of Sport 2 (Schmidt, Barresi)