Sam Warburton's side clung on to secure a 23-21 victory in an absorbing Test match - one which ebbed and flowed throughout.
North crossed for one of the all-time great Lions scores and although Cuthbert added a second after half time, the Lions could have lost the match right at the death.
New Zealand referee Chris Pollock penalised the Lions at the breakdown throughout the 80 minutes and with the last kick of the game, Kurtley Beale had the chance to win it for Australia.
The replacement back slipped on his approach to the ball though, the kick went wide and the Lions went 1-0 up in the three-match series.
The narrow success secured an eighth successive Test victory for the Lions in Brisbane but Warren Gatland would arguably have been disappointed not to have won by more. That they did so at all was the most important point though, the Australians refusing to give up despite seeing their backline crippled with injuries.
Israel Folau showed his finishing qualities to cross for a brace of tries and although Beale and James O'Connor booted 11 points between them, missed kicks came back to haunt the Wallabies.
Robbie Deans' men, already without the likes of David Pocock and Scott Higginbotham, were dealt a blow within 40 seconds, debutant Christian Leali'ifano carried off with a worrying looking injury after attempting a tackle on Jonathan Davies.
Leali'ifano was replaced by Pat McCabe but in the early stages, Pollock's interpretation of the breakdown was plain to see as he twice penalised Brian O'Driscoll at the contact area. Luckily for the Lions, O'Connor was wide from the tee on both occasions.
If the missed kicks had served as a warning, the Lions didn't learn and they soon found themselves behind when Folau crossed for his first try of the day. It was some way to mark his first Test start but the score was all about the creativity of scrum half Will Genia.
After the Lions conceded a penalty inside the Australia 22, Genia took a quick tap, broke away from the covering defence and left Mike Phillips trailing in his wake. A chip through was then collected by Folau who crossed in the corner. O'Connor was accurate with the conversion and Australia led 7-0.
At this stage, the Lions were struggling to get into the game but a penalty from Halfpenny settled their nerves before North showed the world just how dangerous he can be.
After collecting a poor clearance kick from Berrick Barnes, North ran the ball back with intent, his immense power taking him past the covering tackles of Barnes and Genia to cross in the corner. It was a superb individual try and with Halfpenny effortlessly converting, the Lions were now ahead by three.
The score seemed to give the Lions confidence, North seeing another score ruled out by the TMO before a second Halfpenny penalty pushed Gatland's side further clear.
In a hugely physical encounter, both sides possessed plenty of threats but it was North and Folau who were stealing the show. No sooner had the Welsh winger scored than he watched his opposite number add a second, Folau stepping Jonny Sexton all too easily on his way to the line.
O'Connor missed the conversion before Australia suffered another injury, Barnes stretchered off after a sickening clash of heads with Folau. Beale was his replacement and although Halfpenny uncharacteristically missed a penalty with the last kick of the half, the Lions led 13-12 at the break.
Injuries had forced Deans to play his hand early and when McCabe was the next into the medical room, it meant openside Michael Hooper moved into the centre. It was far from ideal from an Australian point of view, the Lions taking advantage almost immediately.
Out of position, Sexton made an intelligent call to test Hooper's midfield defence, finding Cuthbert on an inside ball. It worked a treat, the Welsh winger charging through unopposed to cross for a score which Halfpenny again converted.
Despite trailing by eight points and with a flanker playing in the backs, Australia refused to give up and with the Lions falling foul of Pollock's whistle, a penalty apiece from O'Connor and Beale brought the home side right back into things.
Jamie Heaslip was particularly unlucky to have been penalised for the second of those kicks but Halfpenny replied with his third penalty to settle a few nerves.
Those anxieties soon reappeared though and Beale had two late shots at goal to win it. It was heart-in-mouth time for supporters of the men in red but neither kick was successful and the Lions held on.
Tries: Folau (2); Conversions: O'Connor; Penalties: O'Connor, Beale (2)
BRITISH & IRISH LIONS 23
Tries: North, Cuthbert; Conversions: Halfpenny (2); Penalties: Halfpenny (3)