Glory looked on course for a comfortable home victory when Dino Dulbic put them in front against the winless Wanderers with a powerful header.
The hosts had to wait until the 77th minute to double their advantage through Scott Jamieson, but were then forced to endure a nervy finish.
Perhaps suffering a hangover from their AFC Champions League-winning exploits, Western Sydney have failed to register a single point in the Australian top flight this season.
However, when Jaushua Sotirio pulled one back with five minutes to play, Tony Popovic's side had hope of snatching an unlikely point.
But Perth held on to take their points tally to 15 from six A-League outings this term.
Earlier on Saturday, Adelaide United had moved top temporarily thanks to Cirio's dramatic injury-time winner against Wellington Phoenix.
Phoenix went ahead early on at Coopers Stadium courtesy of Dutchman Roly Bonevacia's first goal for the club, but Nigel Boogard levelled before half-time.
Adelaide piled on the pressure after the break, though, and finally got their reward in the 94th minute when Spanish forward Cirio struck.
The day's other A-League fixture saw Sydney draw 0-0 at home to Melbourne Victory.
