It means the tourists trail the series 2-0 heading into the third Test in Perth on 14 December.
England had resumed on 176-4 with hopes of chasing down their victory target of 354 after fighting back on days three and four.
Their offensive stalled, however, after night watchman Chris Woakes was dismissed by Josh Hazlewood on the second ball of the day.
England skipper Joe Root, on 67 runs, followed him back to the dressing room at the Adelaide Oval soon after, caught behind in the Australian paceman's next over without adding to his overnight score.
Moeen Ali succumbed to off-spinner Nathan Lyon for the second time in the match and was trapped LBW.
Any hope Craig Overton brought to the crease after his unbeaten 41 runs in the first innings was soon drained after he was removed by Mitchell Starc with the second new ball.
Stuart Broad and Jonny Bairstow also fell victim to the left-arm fast bowler who went on to complete the eighth five-wicket haul of his Test career and see England all out for 233.
"We always had faith in the team, the belief," Australia captain Steve Smith said. "It happened really quickly. I'm pleased for the boys. I thought they came out and showed some really good character this morning."
A frustrated Root admitted his side needed to perform better if they were to salvage their fading Ashes hopes.
"It's very frustrating, I thought the way we responded in the second innings with the ball and then the fight and resolve we showed last night with the bat was outstanding.
"It would have been nice to be two or three down going into today, but, credit where credit's due, they bowled very well up front and we just have to be better."
Root's men need a draw in the five-Test series to keep the Ashes, but England have never come from 2-0 down to draw or win a series against Australia.