The tourists led 19-10 at the break thanks to a first minute Alex Corbisiero try and the boot of Leigh Halfpenny (conversion and three penalties).
However, James O' Connor gave Australia hope, with a try just before half-time.
Early in the second half the hosts reduced the gap between them and The Lions to just three points.
But instead of folding, Warren Gatland's men pulled away with tries from Jonathan Sexton, George North and Jamie Roberts.
That handed The Lions a 2-1 series win, their first since defeating South Africa 2-1 in 1997.
