England resumed at 176-4 on Wednesday, needing a further 178 to achieve a national record fourth-innings run chase for a win that would level the Ashes series at 1-1.
That scenario relied heavily on England skipper Joe Root, on 67, posting a big score on day five at the Adelaide Oval.
But after watching night-watchman Chris Woakes (5) feather a catch off Hazlewood's bowling to wicketkeeper Tim Paine on the second ball of the day, Root was also out caught behind in the Australian paceman's next over without adding to his overnight score as England slumped to 177-6.
Nathan Lyon dismissed Moeen Ali and Starc (5-88) returned with the new ball to remove Craig Overton, Stuart Broad and complete a five-wicket haul when he bowled Jonny Bairstow for 36.
The third Test starts on December 14 in Perth, where England are in danger of surrendering the urn after winning the Ashes on home soil in 2015. Australia swept the Ashes 5-0 the last time the series was played Down Under.