The Lions have scored their third try, through George North.
It was created by Leigh Halfpenny, who caught a high ball, beat two Australian tacklers and drew a third, feeding North to finish in the corner.
Halfpenny couldn't add the extras from out wide, but the Lions lead by 18 points.
A try from fly-half Jonathan Sexton - and another Leigh Halfpenny conversion - put the Lions clear again heading into the final quarter.
Sexton touched down after receiving the final pass from Halfpenny in a slick Lions move in the left corner.
Australia had closed the gap to just three points early in the second half through two Christian Leali'ifano penalties.
Halfpenny added another penalty of his own.
Australian fly-half James O'Connor finished a jinking run with a try in the very last play of the first half, to narrow the scoreline to 19-10.
The Lions lead at the break thanks to Alex Corbisiero's early try, and 14 points from the boot of Leigh Halfpenny.
Australia could have been ahead, but turned down five opportunities to kick penalties for goal in the first period.
Halfpenny kicked another nerveless three points, after the Lions earned a further penalty at the scrum.
When the Australian forwards couldn't live with the Lions at scrum time yet again, prop Ben Alexander was sent to the sin bin.
Halfpenny again made no mistake, to extend the Lions' lead to 19-3 after 26 minutes.
Corbisiero's try came after scrum half Mike Phillips took a free-kick quickly, and fed Tommy Bowe on the right wing. The ball came back in field for the England front row to touch down near the posts.
Cardiff Blues full-back Leigh Halfpenny has since kicked two penalties, to one from Australian centre Christian Leali'ifano.The Lions lead 13-3 after 13 minutes.
Prop Alex Corbisiero touched down after just two minutes, to put the Lions ahead against Australia in the series decider.
Full-back Leigh Halfpenny kicked the conversion from in front of the posts, to put the Lions 7-0 ahead.
The British and Irish Lions will go for Test series glory against Australia at Sydney's ANZ Stadium later today. It is 16 years since the Lions won a Test series, and the pressure is on for them to deliver.
The Lions have lost skipper Sam Warburton through a hamstring injury, while 2005 leader Brian O'Driscoll was controversially dropped by head coach Warren Gatland. Alun Wyn Jones takes over as captain of a predominately Welsh side.
The Lions can boast many big-game players and many seasoned internationals in their ranks, but none of them will have experienced the type of pressure that awaits at Sydney's ANZ Stadium. It will call for cool heads, calmness when the heat comes on and clear thinking amid an intense environment.
The British and Irish Lions coach Warren Gatland said the "head overruled the heart" on many of his selection decisions for the final test against Australia in Sydney.
Former Lions captain Brian O'Driscoll said he was "totally gutted" at being left out of the squad for the decisive match this Saturday.
Gatland said: "Picking this team was not easy and ultimately with several players available after recovering from injury the head overruled the heart in many selection decisions.
"Brian O'Driscoll is a great player and has had a wonderful career but for the final test in Sydney, we just felt Jamie Roberts' presence offered us something more".
Former Lions captain Brian O'Driscoll has said he is "totally gutted" at being left out of the squad for the final and decisive Test against Australia: