The hosts, who started the day on 326-6, were bowled out for 375 runs following a century from Ian Bell.
That set Australia a target of 311 to land the first blow in the series, but a devastating spell with the ball from England after tea saw the visitors post a score of 174-6 at the end of play.
Stuart Broad, starting on 47, quickly reached his 50 before Bell added the two runs that he needed to become the first player of the series to hit a century.
However, Watson fell to Stuart Broad soon after lunch following a knock of 46 for the opener.
The visitors looked to have steadied themselves as tea was approaching, but Ed Cowan was bowled by Joe Root with 14 runs to his name.
Rodgers was the first to go after tea, hitting 52 before he was caught by Bell off the bowling of Anderson.
That preceded a dramatic collapse from the Aussies as they went from 161-3 to 164-6.
England edged closer to victory with the very next ball as Phillip Hughes failed to get out of the way of another tricky Swann delivery.
Brad Haddin (11 not out) and Ashton Agar (one not out) will start at the crease on the final day.