Glory leapfrogs Melbourne Victory and sits a point clear of its closest rival after a stunning equaliser from Nebojsa Marinkovic eight minutes into the second half.
The result also gave the Reds third place outright, a point ahead of Wellington Phoenix.
United opened the match the better of the two sides, taking a deserved lead on 11 minutes through Pablo Sanchez.
Tarek Elrich split the Perth defence and threaded a neat pass for Sanchez, who side-stepped his opponent and superbly dinked the ball over Danny Vukovic in the Glory goal.
The Reds could have added a second when Osama Malik rose highest to meet Marcelo Carrusca's free kick, but the defender's header was directed straight at Vukovic.
Two minutes later, Sergio Cirio squandered another golden opportunity to extend United's lead, scuffing his shot from just five metres and missing the vacant net after some great build up play and cross from Michael Marrone.
The visitor stepped up the pace in the second half and United appeared to be feeling the pressure before the Glory eventually drew level in the 53rd minute.
Marinkovic collected the ball on the edge of the area and picked out the top corner of Eugene Galekovic's goal with a stunning 20-metre curling effort that left the Reds keeper no chance.
Perth continued to apply the heat before United had two chances in quick succession.
Substitute Awer Mabil stung the palms of Vukovic from just outside the area after a swift counter attack before Carrusca attempted an audacious first-time chip that the Glory keeper expertly tipped over the bar.
United had a shout for a penalty in the 84th minute when Craig Goodwin whipped in a cross for Bruce Djite, only for the striker to seemingly be brought down as he charged into the box. He recovered to see his close-range shot denied again by Vukovic.
Glory coach Kenny Lowe said his side created enough chances to snatch all three points.
Perth was outplayed for most of the first half and Lowe said he tinkered with tactics at the break.
As a result he saw a much improved second half from his side.
Lowe admitted he is no doubt the happier of the two coaches and said the point on the road against the Reds was well-deserved.
"I can see why he (Adelaide coach Josep Gombau) is a bit upset but, second half we created enough chances to maybe win it," Lowe said.
"So did they, but that's just the nature of it - tough place to come, they're a super team and play some great football.
"We hung in there and got something out of it."
Lowe said the dressing room was upbeat at the break and the half time score gave his troops plenty of confidence heading into the second half.
"At half time they weren't down, there was a lot of optimism there," Lowe said.
"They know it might possibly should've been more than one (goal).
"When it's one (1-0) you've got a chance, there's always hope."
Reds coach Josep Gombau said he was satisfied with United's performance and thought his team deserved maximum points despite wasting numerous opportunities in the first half.
The Spaniard said he isn't concerned his side failed to kill-off the contest for the second consecutive match and remains confident the Reds will continue to play well and win matches.
"We did a good game, this is what I said to the players," Gombau said.
"I think that we deserve to get the three points."
"I am not worried honestly.
"I think that playing like this we will win games."
