With no frontline spinner in the side, Webster claimed 3-51 on day four as Jacob Bethell's century took England to 8-302 and a 119-run lead at stumps.
The Tasmanian's wickets will add more weight to his chances of replacing the under-pressure Cameron Green as Australia's first-choice allrounder, after also reaching 71 with the bat on Wednesday.
The match situation now leaves Australia with a potentially tricky fourth-innings chase amid intrigue over whether Usman Khawaja will open the batting in his final Test innings.
On a wicket offering turn and inconsistent bounce, an Australian victory is by no means a certainty.
But before Webster's spin entered the fray, it looked like the hosts could be facing a more difficult chase with the tourists leading by 36 runs with seven wickets in hand.
At that point Australia were showing signs of fatigue, with Green getting in the way of a catch, Marnus Labuschagne gifting four overthrows and five penalty runs awarded after a ball went through Alex Carey's gloves and into a helmet.
Webster has insisted he does not have a point to prove after being disappointed to be left out for the first four Tests of this series, but his case for a permanent recall now looks as strong as ever.
Having bowled finger spin for the majority of his career, Webster barely practises it any more with his seamers the priority.
As evidenced on Wednesday though, his height means his tweakers can prove as tricky in Australia as they might in different parts of Asia.
The 32-year-old landed a double-blow on England midway through the final session, first breaking a 102-run fourth-wicket stand between Bethell (142no) and Harry Brook.
And after spinning a ball back from outside off to trap Brook lbw on 42, he then had a hard-swinging Will Jacks well caught by a diving Green in the deep two balls later.
Webster later had an injured Ben Stokes caught at slip later after the England captain limped from the field with a groin complaint while bowling in the morning.
"If you told me I was going to take three with off spin I would probably take that," Webster said.
"It's more muscle memory. I bowled a lot of it in my early 20s and teens before I made the switch.
"They are always on tap if ever I need them, and I was lucky enough to get a bit rough outside off and got a few to bite."
Scott Boland (2-34) had Brydon Carse caught at slip late in the day, after earlier trapping Joe Root lbw during a superb eight-over spell.
As much as Webster has been unlucky to be overlooked by Australia this summer, so too has Bethell for England after sitting behind the out-of-sorts Ollie Pope for the first three Tests.
The 22-year-old still didn't get the support he needed on Wednesday, while becoming just the third player in 132 years to score his maiden first-class century in an Ashes Test.
As wickets fell around him Bethell looked as comfortable as any English batter this summer, playing with purpose through the offside with a number of powerful cuts and square drives.
The left-hander then brought up his hundred by using his feet to Webster and hitting the spinner to the long-on boundary.
"Ideally we'd be a couple of wickets less," Bethell said.
"But 119 ahead, I think we've got a chance if we get up to 180 or 190 potentially."
Bethell's century marked the fourth of the match, following Joe Root's in England's first-innings 384, before Travis Head and Steve Smith both reached three figures in Australia's reply of 567.
Stokes will be unable to bowl on day five, leaving England spinner Jacks as the key man.
If he was to perform it would raise further questions over Australia's decision not to play Todd Murphy and go without a frontline spinner at the SCG for the first time in 138 years.
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