At the end of day four at the Sydney Cricket Ground, England had reached 302-8 with Bethell imperious on 142 not out and number 10 Matthew Potts yet to score.
England are desperate for another morale-boosting win after victory in the previous Test at Melbourne. Australia lead 3-1 and have already retained the Ashes.
It was the 22-year-old Bethell who kept their faint hopes alive after Zak Crawley fell early and first-innnings centurion Joe Root departed cheaply.
He powered to his hundred off 162 balls by smashing a lofted drive over midwicket for four.
"It's pretty special, it's not really sunk in yet. I haven't really had time to kind of reflect on it," Bethell told reporters.
Bethell had his parents in the stands and raised his bat towards them as he passed the milestone.
"To do that and have the family there was very special," said the left-hander.
"My dad was a cricketer so he knows how it feels to be out there batting, but I don't think that stops him from getting any less nervous."
It was not just Bethell's maiden Test ton but the only one he has scored in red-ball cricket, with his previous best 96 against New Zealand.
Bethell was recalled for Melbourne to replace the under-performing Ollie Pope, scoring a gritty 40 in difficult batting conditions.
He proved to be England's rock, but soon saw wickets tumbling at the other end as Harry Brook was trapped lbw for 42 and Will Jacks caught without scoring in the space of three balls from part-time off-spinner Beau Webster.
"I certainly didn't think I would (impact) with spin this game," said Webster, who is more usually employed as a seamer, but returned figures of 3-51.
Webster believed the pitch had begun to take turn. "There was a bit of wear and tear there from Starcy's footmarks," he said. "And we managed to make the most of it in that session."
A mix-up saw Jamie Smith run out for 24 just as he and Bethell were threatening to take the game away from Australia with a sixth-wicket partnership that had reached 45.
Ben Stokes then came in at number eight with the score 264-6 after limping off with groin injury while bowling, but the captain lasted only five balls.
Australia were dismissed for 567 before lunch on the back of Travis Head's 163 and 138 from skipper Steve Smith, a first-innings lead of 183 runs.
England needed a solid start, but chief tormentor Mitchell Starc once again conjured up a first-over breakthrough -- his 29th wicket for the series.
A beautiful inswinger was misjudged by Crawley who shouldered arms as it thudded into his pad and he was out lbw for one.
