The Lions' victory was built on a dominant opening half when they exploded to a 19-3 lead, before the Wallabies pulled back to trail 19-10 at half-time.
But despite having their lead reduced to three points, the tourists stormed home with three second-half tries before a record 83 702 crowd.
The Lions overpowered the Wallabies in the final quarter of the match to finish four tries to one victors, with man of the match Leigh Halfpenny kicking five penalties and three conversions for an individual record haul of 21 points - beating Jonny Wilkinson's 18 points in 2001.
It was the most points the Lions have scored in a Test win in Australia, eclipsing their 31-0 victory in Brisbane in 1966.
It was a triumph for coach Warren Gatland, who caused an uproar in Ireland with the dropping of Irish Lions legend Brian O'Driscoll and fielding a record-equalling 10 Welshmen in the starting XV.
But the Lions responded to the pressure of the big-match occasion with their three tries inside the final 25 minutes to win an epic series after taking out the opening game in Brisbane 23-21 a fortnight ago.
The large Lions contingent of supporters celebrated in the final minutes by singing 'Sweet Chariot' in salute to their team's first series win since South Africa in 1997.
The Lions had a terrific start after Wallaby lock Kane Douglas knocked on from the kickoff and the tourists went straight on the attack from a scrum win.
Prop Alex Corbisiero powered over for the opening try next to the uprights for Halfpenny to convert and give the tourists a 7-0 lead after two minutes.
George Smith, recalled after four years out of Test rugby, had a sickening clash of heads with Lions hooker Richard Hibbard and was led groggily from the field to be assessed by team medics.
Halfpenny consolidated the Lions' lead with a penalty from near halfway to lead 10-0 but Christian Lealiifano put Australia's first points on the board with a penalty goal as Smith returned to the field.
French referee Romain Poite penalised the Australian scrum and Halfpenny made the Wallabies pay with another long-range penalty to stretch the lead back to 10 points after 13 minutes.
The Lions scrum was dominating and Poite penalised the Wallabies again, giving Halfpenny an easy shot to extend the lead to 16-3 in an emphatic start to the Test.
Poite cracked down in the scrum and sent Wallaby prop Ben Alexander to the sin bin for repeated infringements, giving Halfpenny another penalty goal and a 19-3 lead after 25 minutes.
It was ninth player the former policeman Poite had sent to the sin-bin in his last eight Tests.
Everything continued to go against the Wallabies when their wing ace Israel Folau tweaked his hamstring in running back after a kick and he made way for debutant Jesse Mogg.
But the Wallabies had the Lions under pressure on their try-line as the halftime siren sounded, and flyhalf James O'Connor gave the home side hope when he jinked past Sexton and carried Mike Phillips's tackle for a converted try and 19-10 scoreline at the break.
And Australia edged closer after the resumption with the Lions penalised in a maul and Lealiifano kicking a penalty for the home side to move to within six points.
The Lions were put to the test when Poite whistled another penalty within kicking range off a maul, and the Wallabies were down by just three points six minutes into the second half.
However, another dominant Lions scrum gave Halfpenny his sixth successful kick to give the tourists some breathing space at 22-16 after 51 minutes.
Sexton kicked the Lions away when he scored a converted try off a Halfpenny pass in the 56th minute, and the tourists were in the box seat at 29-16.
The winning break came when Halfpenny fielded Genia's clearing kick and launched a counter attack, beating the tackles of Genia and Joe Tomane, to send winger George North racing over for an unassailable 34-16 lead.
And Jamie Roberts, a controversial replacement for the dropped O'Driscoll, vindicated Gatland's selection when he arrowed through for the final try with 11 minutes to go.
Try: James O’Connor.
Conversion: Christian Lealiifano
Penalties: Lealiifano (3)
British and Irish Lions:
Tries: Alex Corbisiero, Jonathan Sexton, George North, Jamie Roberts.
Conversions: Leigh Halfpenny (3)
Penalties: Halfpenny (5)
15 Kurtley Beale, 14 Israel Folau, 13 Adam Ashley-Cooper, 12 Christian Leali'ifano, 11 Joe Tomane, 10 James O'Connor, 9 Will Genia, 8 Wycliff Palu, 7 George Smith, 6 Ben Mowen, 5 James Horwill (captain), 4 Kane Douglas, 3 Ben Alexander, 2 Stephen Moore, 1 Benn Robinson
Substitutes: 16 Saia Fainga'a, 17 James Slipper, 18 Sekope Kepu, 19 Rob Simmons, 20 Ben McCalman, 21 Michael Hooper, 22 Nick Phipps, 23 Jesse Mogg
British and Irish Lions:
15 Leigh Halfpenny, 14 Tommy Bowe, 13 Jonathan Davies, 12 Jamie Roberts, 11 George North,10 Jonathan Sexton, 9 Mike Phillips, 8 Toby Faletau, 7 Sean O'Brien, 6 Dan Lydiate, 5 Geoff Parling, 4 Alun Wyn Jones (captain), 3 Adam Jones, 2 Richard Hibbard, 1 Alex Corbisiero
Substitutes: 16 Tom Youngs, 17 Mako Vunipola, 18 Dan Cole, 19 Richie Gray, 20 Justin Tipuric, 21 Conor Murray, 22 Owen Farrell, 23 Manu Tuilagi