The match in County Durham was intriguingly poised at lunch with the tourists on 11 for no loss and England already one review down.
The home team, 2-0 up in the five-match series, went to the Decision Review System when umpire Tony Hill rejected an lbw shout by Stuart Broad against first-innings centurion Chris Rogers.
Hill was vindicated when television replays showed the ball had pitched outside leg stump.
Earlier, Harris took seven for 117 to surpass his previous test best of six for 47 achieved against the same opposition in Perth in December 2010.
In-form Ian Bell resumed on his overnight score of 105 and quickly took his run tally for the series to 500 before Harris bowled him for 113 with an inswinger that stayed low.
Matt Prior followed one ball later for a golden duck after another perfect inswinger from Harris deflected off his elbow and on to the stumps.
The Australian paceman then removed Broad for 13 with a vicious bouncer that the all-rounder gloved to Steve Smith in the gully.
Nightwatchman Tim Bresnan went on to the counter attack with a bright and breezy innings before Harris had him caught and bowled for 45.
Nathan Lyon removed last man James Anderson for a duck, leaving Graeme Swann unbeaten on 30.
Cricket Australia said all-rounder Shane Watson, who could not complete his seventh over on Sunday because of pain in his right hip and groin, would bat as required before having his injury reassessed at the end of the match.
(Editing by Ed Osmond)
