In the 80th minute, Perth Glory broke through when Susan Phonsongkham curled the ball from a corner, and it struck the outstretched leg of Canberra defender Hayley Taylor-Young.
The ball soared into the back of the net and the Sam Kerr Football Centre crowd erupted on Saturday.
But before the corner Phonsongkham fired a shot towards goal that was palmed away by keeper Sally James.
United players were left screaming at referee Isabella Mossin for a handball, after the 24-year-old took a touch with her upper body before shooting, but Mossin waved away the call.
"She (Mossin) said she didn't see anything, but then her linesman said they saw it. When I asked the linesman, they said 'the sun's in my eyes and I can't see anything'," Heyman told Paramount+.
"It's a plain lie straight to the face, no excuses about it. She just wasn't really watching where the ball was.
"This is the A-Leagues, and this is the quality that sometimes we get. It's a little bit frustrating."
The win moves Perth to seventh on the ladder, trailing Melbourne Victory by goal difference with one game remaining before the league breaks for the Women's Asian Cup beginning on March 1.
Glory dominated large parts of the contest as Canberra were scrappy, posting just two shots on target and converting 60.3 per cent of their passes.
A victory would have catapulted them into top spot on the ladder, but instead Canberra (24) sit behind Melbourne City (26 points) and Wellington (24), who both have at least a game in hand.
"I'm pretty relieved, to be honest. It was hard out there for large parts of that game, so it's nice to come away with three points," Perth captain Isobel Dalton said.
"We take a lot of confidence from that today. We have to be proud of this performance."
Canberra had the first proper look at goal when midfielder Bethany Gordon lofted a free kick towards the net, but Teresa Morrissey made a comfortable save.
The game became a midfield grind as both teams battled for possession. Perth's Rola Badawiya nearly rewarded her side's efforts when she curled the ball towards the bottom left corner, but James denied her.
The Glory almost drew first blood in the 59th minute when Phonsongkham hit the ball from a volley, but James's outstretched palm pushed the ball over the crossbar.
After the controversial goal, Canberra desperately tried to find an equaliser, but Perth held strong to win.
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