Hazlewood trapped Root leg before wicket for 51 to send the tourists tumbling to 119-5 at the interval, a lead of 93.
At lunch, Moeen Ali was on unbeaten on 26 with wicketkeeper Jonny Bairstow on 1.
Root's dismissal was a significant blow for England's chances of building a lead for Australia to chase in the fourth innings with five sessions of play remaining.
Root looked in good touch with his 160-minute stay before Hazlewood rapped him on the pad to the jubilation of the Australian team.
Earlier, off-spinner Nathan Lyon removed opener Mark Stoneman to a low catch at slip by Peter Handscomb for 27 and followed up with the wicket of Dawid Malan for four shortly after drinks.
At lunch, Hazlewood had taken three wickets for 38 and Lyon 2 for 36.
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