In-form City were on course to upset the odds and record their third straight top-flight win when Aaron Mooy put them ahead shortly after the break.
Mooy collected a pass from David Williams and finished well to give the hosts a deserved lead in the 48th minute.
However, table-topping Perth levelled just after the hour mark, when experienced former Hull City midfielder Garcia stooped to head Scott Jamieson's free-kick into the ground.
The ball bounced up and over wrong-footed City keeper Tando Velaphi to put Glory back on level terms.
And the league leaders managed to hold firm in the face of strong pressure from City to claim a 1-1 draw.
In the A-League's other Boxing Day fixture, Adelaide United moved up to second place thanks to a second-half blitz that gave them a 3-0 win at Sydney FC.
Following a scrappy first half of few chances, Adelaide moved through the gears following the interval and Bruce Djite put them in front with 67 minutes gone.
Goals in the final 10 minutes from Pablo Sanchez and Awer Mabil then added gloss to the scoreline as Adelaide leapfrogged Melbourne Victory into second, with Kevin Muscat's men set to host Newcastle Jets on Saturday.
