The start of the second session was delayed for an hour by rain, keeping England skipper Joe Root, who had made 13 not out, and injured all-rounder Ben Stokes, who was unbeaten on 16, in the dressing room.
The revised playing conditions left Australian needing seven wickets in 56 overs for victory, providing no more rain materialises.
The tourists had resumed the day on 13 without loss chasing an unlikely victory target of 388 and Zak Crawley led the resistance at the Sydney Cricket Ground with an authoritative 77 from a hundred balls.
His fellow opener Haseeb Hameed looked less assured at the other end and had already been dropped when he prodded at a Scott Boland delivery and departed for nine to give the paceman his 12th wicket in four Test innings.
Dawid Malan had made four when he stepped back to a full-length Nathan Lyon delivery, which skidded on and rattled into the off stump.
Crawley had hit 13 fours in his fifth Test half century when Australia all-rounder Cameron Green got under his bat with a yorker.
The batsman reviewed the lbw decision but the DRS video showed the ball had landed plumb on Crawley's toe as it headed towards the stumps.
Australia lead the series 3-0 after big wins in Brisbane, Adelaide and Melbourne. The fifth Test will be a day-night match in Hobart starting on Jan 14.