The tourists slipped to 6-124 at the end of day two's second session in Hobart, having reduced Australia's lead to 179 runs.
Debutant Sam Billings was unbeaten on 19, while Chris Woakes was five not out after being dropped on zero and five.
Cummins led from the front, removing Root and Dawid Malan in quick succession during a collapse of 3-7 that also featured the key scalp of Ben Stokes.
Ollie Pope steadied briefly before succumbing to Scott Boland on 14, punching the bat in anger after edging a wide ball to wicketkeeper Alex Carey.
Cummins failed to review two caught-behind shouts, with replays indicating they represented missed chances to dismiss Malan and Rory Burns.
But such was Australia's dominance, and their skipper's capacity to create pressure despite the shine wearing off the pink ball, that neither moment really mattered.
Burns was needlessly run out for a duck by Marnus Labuschagne's direct hit, while fellow opener Zak Crawley was undone on 18 when Travis Head's juggled short-leg catch handed Cummins the first of his three wickets.
Root and Malan batted resolutely in their 49-run stand after being asked to rebuild for the umpteenth time this summer.
Malan did the hard work then threw his wicket away, providing a leg-side catch for Carey.
The lbw dismissal of Root, whose 34 currently represents England's top score, is more likely to feature on Cummins' end-of-summer highlight reel.
The delivery seamed in and pinned Root, deep in the crease.
Cummins celebrated with gusto, punching the air, while Root shook his head and trudged off in scenes reflective of the manner in which Australia eased to an unassailable 3-0 lead in the five-Test series.
Stokes managed one boundary, finding a gap between leg slip and Carey, before falling victim to a sensational catch from Nathan Lyon at point.
Lyon is yet to bowl, presumably because Cummins is waiting for the ball to soften up, but the offspinner's free-wheeling knock of 31 lifted Australia to a first-innings total of 303.
Lyon hoiked three bouncers from express paceman Mark Wood over the leg-side fence, including one thunderous blow that sent the pink ball whizzing out of the ground.
Lyon and Boland's final-wicket stand of 23 boosted Australia's score past 300, notably higher than what England have managed in any of their eight previous innings on the tour.
The hosts resumed at 6-241 on Saturday, having collapsed to 3-12 on day one but then seized control thanks to Head's century.