Glory lengthened its winless run to five matches, looking anything but the league leader.
Kenny Lowe's side didn't have a shot on target all match, leaving much of the attacking to finals-chasing City.
Josh Kennedy and Aaron Mooy impressed for City but found the sticky conditions too tough to break the deadlock.
City controlled the first 20 minutes but couldn't put the ball in the net.
Patrick Kisnorbo headed a corner downwards instead of towards goal, while a strong run and cross from lively City debutant Kew Jaliens gave Kennedy a sight of goal on 14 minutes.
The former Socceroos striker toed his effort over the bar as Josh Melling peppered Danny Vukovic's goal.
With temperatures stuck above 30C, the match degenerated.
Andy Keogh and Josh Risdon, who were detained by police in the aftermath of Glory's 1-1 draw with Adelaide, shook off their hangovers and looked Perth's most likely danger men.
But the league leader couldn't fashion a shot on goal during a first half that petered out in the heat.
Jaliens missed a header of his own from Mooy's fine delivery five minutes after the break.
Worse for City, Melling limped off for City on the hour mark with a suspected knee injury.
Glory's injury news was much better, with Mitch Nichols making a return as a substitute after 10 weeks out with a hamstring strain.
Perth wasn't without chances but was largely restricted to counter-punching.
Keogh sent two second-half headers wide of the bar in a match that seemed destined to end nil-all, before being sent clear in injury-time.
Nichols's fancy footwork found the Irishman but Tando Velaphi was quick to rush out and preserve his clean sheet.
City stays stuck in seventh with the draw, the third consecutive match in which John van't Schip's team has failed to score.
Van't Schip said it was closer to two points dropped rather than a point earned.
"They were more happy with the point than we were," he said.
"I'm happy with the clean sheet, if you have to be happy about something."
The Dutchman said his side could have played against 10 men after Denis Kramar's late and blatant foul on Kennedy after 77 minutes.
Lowe, who also took a 1-1 draw back across the Nullabor from his previous visit to City, said the match wasn't much of a spectacle.
"I'm glad I didn't have to pay to watch it," he said.
"We came and did what we came to do: we're quite happy with a point."
