The hosts trailed 13-6 at the break after a try by Matt Toomua and three kicks by Quade Cooper, with England’s points coming from the boot of Owen Farrell.
But quick-thinking by Chris Robshaw saw the England captain swoop on Mako Vunipola’s charge down and over the line from close range. It was his first international try and, converted by Farrell, brought England level after 50 minutes.
Farrell then raced through a gap in the Wallabies defence to give England the lead eight minutes later, before dusting himself down to convert his own try.
England withstood some late Australia pressure to ensure Australia remained pointless in the second period and secure the victory.
Tigers forwards Dan Cole and Tom Youngs started the game and Ben Youngs and Toby Flood made an impression from the bench in the second half, Flood replacing his former Tigers teammate Billty Twelvetrees at centre.
England play Marcos Ayerza’s Argentina at Twickenham next Saturday in the second of their three autumn fixtures before they face the All Blacks the following weekend.
