Nicholas Pennington's 66th-minute tap-in and Charbel Shamoon's 88th-minute strike from outside the box ensured the visitors took all three points from Sunday's A-League Men match.
The result marked the Glory's second win on the trot - both on the road - after they beat Newcastle 2-1 a week earlier.
Griffiths was thrust into the hot seat after David Zdrilic was sacked following Perth's 4-0 loss to Melbourne City in round two.
Although the Glory suffered losses in Griffiths' first two games in charge, the team started showing more attacking enterprise.
More things started to click in the win over the Jets, and Perth produced their best display of the season in the victory over Macarthur, keeping their opponents to just three shots on target.
The Glory unleashed 15 shots for the match, five of which were on target.
Perth hierarchy has expressed their desire to appoint an experienced coach but with Griffiths' cause growing by the week, the 46-year-old former Wanderers and Phoenix assistant could be hard to ignore.
"Since Griffiths has come in, the way he wants to play, everyone's bought into it, and we look a completely different team," Glory star Tom Lawrence said.
"It's not my decision (on the coach), but I think it shows all the players have bought in, everything he wants.
"He's been brilliant. We had that international break to really work on how we wanted to play.
"Last week was a bit of a 'just get through it and win the game'.
"This game we showed our quality, and there's a lot more to come."
The result left both the Glory and Bulls level on seven points after six games, just one point adrift of sixth spot.
There were only two changes to Macarthur's starting line-up from Thursday's 2-1 win over Hong Kong outfit Tai Po FC in the AFC Champions League 2.
Lawrence almost opened the scoring in the ninth minute, only for his rolling strike to bounce off the upright and back into the arms of goalkeeper Filip Kurto.
There had only been a total of three shots on target between the two teams by the time the match rolled past the 60-minute mark, but Lawrence gave the game the spark it needed.
The former Welsh international toed the ball ahead of an opponent during a fast break, raced into the box before squaring the ball for Pennington to tap into an empty net.
One-cap Iraq international Shamoon iced the win in the dying minutes when he found himself in space and cut inside before unleashing his shot into the back of the net, securing his first ALM goal in his 27th appearance.
