Phoenix looked set to be frustrated by a stubborn Adelaide outfit, but Joel Griffiths finally broke the deadlock 11 minutes from time.
Nathan Burns sealed the three points with a second goal shortly after to temporarily move Ernie Merrick's side top of the table on goal difference.
And stuttering Perth blew the chance to immediately restore their three-point advantage as they were held to a 1-1 draw at fellow title-chasers Melbourne Victory.
Glory travelled to AAMI Park without a win in their previous six A-League games, and Daniel Georgievski's leveller on the hour - which cancelled out Andy Keogh's first-half opener - gave the hosts a hard-earned point in an ill-tempered affair.
Perth's frustration was there for all to see, with the visitors receiving nine bookings - including two for Mitchell Nichols, who was dismissed in injury time.
Nichols will now miss next week's crucial top-of-the-table clash with Wellington.
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