Hazlewood snuffed out England's outside hopes of a come-from-behind victory when he dismissed captain Joe Root in the third over of the final day of the first-ever day-night Ashes Test.
Starc mopped up the tail to finish with five for 88.
Holders England now face the prospect of losing the series in next week's third Test at Perth's WACA Ground where they have not won a Test since 1978.
"We always had the faith in the team and the belief," Australia skipper Steve Smith said.
"We thought if we could come out and bowl well this morning, hit our lengths and get one or two wickets then things can happen quickly."
While Root was batting England's hopes lingered of bringing off a record run chase of 345 at Adelaide Oval.
But Hazlewood enticed a bottom edge and wicketkeeper Tim Paine did the rest to the unrestrained glee of the Australian team.
Root left the field on his overnight score of 67 and with him England's hopes vanished.
"Disappointing. I don't think we did ourselves justice," Root said.
"We're still massively in the series. We showed that from the way we played yesterday.
"That has to be the benchmark how we approach the rest of the series."
Nightwatchman Chris Woakes was out to the second ball of the day, caught behind off Hazlewood for five.
Woakes sought a review and while the 'Hot Spot' infrared imaging system could not find anything, the 'Snicko' sound-video technology detected an edge and the umpire's decision was upheld.
Nathan Lyon got the wicket of fellow off-spinner Moeen Ali for the fourth time in the series, leg before wicket as he attempted to sweep when on two.
That left England lurching towards defeat on 188 for seven after 45 minutes of play on the final day.
Craig Overton stayed around for 39 minutes before he fell lbw to a Starc inswinger for seven with the first delivery of the second new ball.
Starc had Stuart Broad caught behind for eight and finished off the Test just before tea bowling Jonny Bairstow for 36.
England entered the final day 178 runs from victory with six wickets in hand after a dramatic momentum switch on Tuesday.
Australia dismissed England for 227 on Monday but decided against sending them back in to bat despite holding a 215-run first innings lead.
The tourists subsequently skittled out the Australians for 138, leaving them with a chance of chasing down a record 354-run victory target in the final five sessions of play.
Skipper Root led a spirited English fightback and relished the challenge of a record run chase with a fighting unbeaten half-century.
The highest winning fourth innings at the Adelaide Oval remains 315 for six by Australia against England in 1902.
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