Australia led 13-6 at halftime after a try from centre Matt Toomua as Farrell missed three of his five penalty attempts after England dominated the scrum but without ever threatening the Wallaby line.
They crossed it five minutes into the second half, though, when captain Robshaw scored his first international try, and added a second soon after when Farrell found a hole and slotted his second conversion.
Australia, seeking a fourth win from five games at Twickenham, pressed in the closing stages but England's defence held firm to condemn the Wallabies to their eighth defeat in 11 tests this year.
England made a nervy start to the match despite taking the first points when Farrell landed his first kick.
Australia quickly levelled through the boot of resurgent Quade Cooper before his England counterpart missed three kicks in quick succession.
A better kick from Farrell saw England regain the lead just before the half-hour mark but within three minutes Australia had completely turned the game around as Cooper combined with Israel Folau for a moment of brilliance.
The ball was then recycled through a patchy Will Genia to Toomua, who dived over the line for the first try of the afternoon. Cooper added the conversion before another penalty in quick succession saw the Wallabies go into the interval with a nice lead in hand.
Australia looked the more likely to extend that lead at the restart as England again struggled to find their rhythm.
But inch-by-inch the hosts worked the way back into the match, cutting out the mistakes and wearing down the increasingly tired and short on confidence Australian team.
Maku Vunipola was very quick to charge down a Genia clearance kick after man-of-the-match Mike Brown and Twickenham debutant Marland Yarde had combined on the left wing.
Captain Robshaw was the first to react to the charge-down, forcing his way over the line to touch down for England.
Farrell added the conversion before taking full advantage on a gaping hole in the Australia defence to record his own try. Australia had a decent call of obstruction against substitute Dylan Hartley but the referee and television match official decided there was “not enough” obstruction to disallow the try and England, once Farrell had converted, had the seven-point lead.
From there it was a case of holding on for Lancaster’s men as Australia had a couple of chances to secure the draw.
The best of those chances came two minutes from time but Benn Robinson knocked on and England cleared before winning three consecutive scrums with penalties to take victory.
England host Argentina and New Zealand at Twickenham in the next two weeks while Australia move on Rome and then Dublin to face Italy and Ireland.
- Sports & Recreation