Root's side only need to draw the series to retain the Ashes, but Australia are heavy favourites to win back the urn and can do so with victory in the third Test match in Perth, starting on 14 December.
Left-arm quick Mitchell Starc mopped up the tail to take five for 88, his eighth five-wicket haul in tests, but Hazlewood will also share the plaudits after his early intervention. Usman Khwaja and Mitchell Starc were the only batsmen to cross 20 runs.
Chris Woakes (5) edged Hazlewood through to Tim Paine off the second ball of the day and was given out, his review failing after Snicko showed a noise but nothing came up on Hot Spot.
"I didn't think it would happen that quickly. Looking forward to Perth, 2-0 up, looking forward to recharging and having another crack", Marsh said after the victory.
Root's innings was always going to be key to England hopes but he was caught behind off a delivery that nicked the bottom of his bat before he could add to his overnight tally of 67 and the last six wickets fell for just 57 runs.
Root, however, simply walked from the field after edging behind. England fought really hard with the ball and last night.
Jonny Bairstow offered some resistance with a well-made 36 before he was bowled by Starc as England were skittled out for 233, 120 runs short of the target. Australia were bowled out for just 138 in their second innings as England's bowlers feasted on seaming conditions under floodlights, especially James Anderson, who claimed five wickets.
"Disappointing", said Root. "I don't think we did ourselves justice".
Only one captain in the long history of test cricket had previously lost a test after opting not to enforce the follow-on and Smith was grateful to his bowlers for getting him out of a hole.