No 8 Billy Vunipola opened the scoring with a try in the fifth minute after England won a close-range lineout and set up a driving maul.
Farrell converted from wide on the left and added four penalty goals in quick time, regularly taking the points as England's forwards dominated the breakdown, possession and territory and forced the Wallabies into errors.
Australia's woes were compounded when Izaia Perese, who replaced a concussed Jordan Petaia, was yellow carded in the 22nd minute after the TMO ruled that he made a deliberate knock-on to shut down a scoring opportunity for England.
But the Australians rallied toward the end of the half, taking an attacking lineout from a penalty. Loosehead prop Angus Bell went close to the tryline before his fellow front rower Tupou burrowed over just before halftime to reduce the margin to 12 at the break.
Another Farrell penalty gave England a 22-7 lead three minutes into the second half before Australia responded with a try to powerful centre Samu Kerevi following an 11-phase attack and a penalty from Noah Lolesio — after his opposing flyhalf Marcus Smith was sin-binned for a deliberate knock-on — to make it 22-17.
The Wallabies created more attacking opportunities but were let down by poor finishing. After repelling Australia's attack, England grabbed crucial points when Farrell extended the cushion to eight with 13 minutes remaining.
Injuries will test the depth of both teams. The Wallabies lost Pataia, Perese, lock Caderyn Neville and prop Scott Sio during the match and England lost influential forwards Maro Itoje and Sam Underhill to injuries.
The result ended Australia's 10-test winning streak at Suncorp Stadium in Brisbane, but extended England coach Eddie Jones' unbeaten run in seven tests at the venue, a stretch which includes his time as head coach of the Wallabies.
- AP