"A lot of people have been asking me that question (if Australia should have enforced the follow-on) ... fortunately we've been able to hang in there and get the result we're after. It doesn't matter any more."
Shaun Marsh was named man of the match for his unbeaten first-innings ton.
Joe Root resumed at 67 on day five, with his side 176-4 and requiring a further 178 runs to complete a record run-chase of 354.
History suggested it wasn't possible - England have never chased down more than 332 to win a Test - but the agitated expression on Smith's face confirmed it was.
Root shaped as the tourists' beacon of hope but the captain was dismissed by Hazlewood without adding to his overnight score, undone by a hint of reverse swing and express pace.
Hazlewood, otherwise below his best in the second Test, almost hit the 150 km/h mark in a fired-up opening spell that also featured the dismissal of nightwatchman Chris Woakes.
Mitchell Starc later had the second new ball hooping, snaring three wickets to finish with figures of 5-88 as Australia turned the screws to spare Smith the ignominy of losing after his declaration on day two.
"We showed a lot of character. The way the senior players stood up in the field especially, to bowl them out for what we did (on day four)," Root said.
"Today was disappointing, I don't think we did ourselves justice ... but we're still massively in the series."
Root and Woakes were both caught behind, with the latter succumbing to the second delivery of the day.
Umpire Aleem Dar pondered Hazlewood's appeal for what seemed like an eternity before giving Woakes his marching orders. Woakes reviewed but the snickometer showed a faint edge.
Root, who swapped barbs with Nathan Lyon, Peter Handscomb and Tim Paine at various flashpoints of a tense Test, nicked off and trudged off during Hazlewood's next over.
"I didn't think it would happen that quickly," Hazlewood said, praising Lyon's lbw dismissal of Moeen Ali.
"There were probably a few nerves (at the start of play but) ... we were pretty confident."
Cameron Bancroft shelled a catch when Craig Overton was on five but England's debutant was trapped lbw by the first delivery with the second new ball - a pinpoint inswinger from Starc.
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