Ulster winger Tommy Bowe started the game and although he didn’t get on the score-sheet he played an important in role in the win and the overall series victory.
The Lions got an early lead prop Alex Corbisiero crossed the line for a second minute try.
Wales fullback Leigh Halfpenny kicked 12 points to put them 19-3 up after 26 minutes.
Australia fought back as James O'Connor scored a try just before half-time and inside centre Christian Leali'ifano kicked the home side to within three points at 19-16 with 25 minutes remaining.
But the Lions, who last won a Test series in 1997 in South Africa, then found the attacking rhythm that they lacked in the second Test defeat.
Fly-half Jonny Sexton, winger George North and centre Jamie Roberts crossed for second-half tries as the Lions ran outcomfortable winners.
It was the Lions first test series win since Jeremy Davidson et al won in South Africa in 1997.
Huge congratulations to Tommy Bowe on being part of rugby history and to Rory Best and Tom Court on their important contributions to the tour.