The Socceroos stay third in Asia's group B with two games remaining but join the second-placed Saudis and leading Japan, who have a game in hand, on 16 points.
The top two teams qualify automatically for Russia, with the third placed side going into a playoff.
Forward Tomi Juric scored a first half brace but the Green Falcons showed courage to equalise twice through goals from Salem Al Dawsari and Mohammed Al Sahlawi.
Australia coach Ange Postecoglou was relieved after his side escaped with the win after a poor first half laden with defensive errors.
"Obviously the three points were important when you look at how tight the group is," he said.
"What we want to do now is go to Japan and knock them off.
"If we do that we don't have to worry about goal difference or points."
Australia travel to Japan in August before closing their campaign at home to lowly Thailand.
Bert van Marwijk's Saudi Arabia face United Arab Emirates away before hosting the Blue Samurai in a blockbuster finale.
Goal difference could ultimately prove crucial and Juric rued sending a late, left-foot chance flying over the bar that would have had his team in line to leapfrog the Saudis into second in qualifying.
The home side also felt aggrieved when winger Mathew Leckie went down in the area in the 70th minute after a heavy challenge from Salman Al Faraj, but Uzbek referee Ravshan Irmatov was unmoved.
The Socceroos were gifted a perfect start by Saudi goalkeeper Yasser Al-Mosailem, who slipped during his clearing kick and sent the ball straight to Juric, who duly hammered it back into the net in the seventh minute.
The Saudis responded well to the setback, and jet-heeled midfielder Al Dawsari crashed through at midfield to slash a low shot past keeper Mat Ryan in the 23rd minute after a neat one-two with striker Al Sahlawi.
The Saudis fell behind again when Leckie burned up the right flank and fired a pin-point cross to Juric, who nodded home his second goal into the left corner.
But a long lofted through-ball sliced open the Australian defence and Al Sahlawi latched onto a header into the area to volley the ball home.
A more aggressive Australia emerged after the break to win control of the midfield, allowing Rogic to step up and launch his thundering strike from just outside the area.
(Reporting by Ian Ransom in Melbourne; Editing by Gareth Jones)
