Sent to the bottom of the league by Western Sydney's draw with Brisbane on Friday, the Phoenix huffed and puffed throughout Saturday's opening half in Wellington, before eventually securing their breakthrough early in the second.
At a third successive corner, Ramy Najjarine swung a 55th-minute delivery to the top of the six-yard box, where Corban Piper rose over teammate Bill Tuiloma and marker Brian Kaltak to fire home a header.
It was two in the 84th minute when, with Trent Ostler off receiving treatment, Glory's calamitous attempts to clear a Matthew Sheridan cross saw the ball bounce off Nicholas Pennington's chest, wrong-foot Matt Sutton and go in for an own-goal.
Remarkably, the three points meant that the Phoenix, despite entering the game last, ended their afternoon closer to the playoff places than the table's foot: going three points clear of the Wanderers and two back of sixth-placed Central Coast.
"We addressed a few things at halftime," Wellington's Alex Rufer told Sky Sport.
"The last few games we've been in the game, and we've not come out well enough in the second half.
"We needed to put them under pressure; we did that, and we deserved the three points."
Staging one of the longest away days in world football, a second-minute Luca Tevere free kick gave the Glory an early look at goal, but the hosts would go on to have 18 touches in their penalty area in the opening stanza compared to one, as well as creating chances for Kazuki Nagasawa and Tuiloma.
Piper's header then finally gave the Phoenix the lead, as well as a level of momentum that allowed them to manage the game out and earn interim coach Chris Greenacre the first win of his tenure.
"A lack of quality on our part," Perth's Scott Wootton told Sky Sport.
"I can't remember the Wellington keeper making a save; that's something we've not been good enough at most of the season.
"Poor from our part."
Winless since January, the Glory fell to second-bottom as a result of the defeat with five games remaining on the season -- memories of their midseason revival under Adam Griffiths fading as they battle to avoid a fourth wooden spoon in five seasons.
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