Captain Paul O'Connell, centre Jonathan Davies, blindside flanker Dan Lydiate and replacement second row Alun Wyn Jones also celebrated scores to hand head coach Warren Gatland a winning start to the 10-match adventure which culminates in a three-Test series with the Qantas Wallabies this month and next.
Gatland's men led by 20 points at the interval thanks to two tries just before and after the half-hour mark, but much of the first period proved fiercely competitive.
Owen Farrell, who finished with 15 points, was heavily involved throughout as he opened the scoring with a 40-metre penalty after just three minutes before being hit with a left hook by Saracens team-mate Schalk Brits moments later.
The Springbok hooker saw yellow for his indiscretion with seven minutes played after the TMO overturned referee Steve Walsh's decision to penalise The Lions on the edge of their own 22.
The tourists were unable to capitalise on the extra man, however, despite going close through wing Sean Maitland. The former Canterbury star chased Glasgow colleague Stuart Hogg's chip ahead only to see Dimitri Yachvili get a boot to the ball just as he was about to gather and score after 11 minutes.
Maitland was again in the thick of the action six minutes later as The Lions showed their class with a flowing backline move from turnover ball 10 metres inside their own half. Davies' fine take and give handed Maitland the chance to skip through Elliot Daly's tackle but a double hit from Jared Payne and Casey Laulala forced a knock on.
With Farrell having been off target with his second attempt at goal a little earlier, the Barbarians weathered the storm and hit back when Daly's 50-metre penalty levelled the scores after a quarter of an hour.
Richie Gray's short burst then almost created a score wide on the right but Hogg couldn't quite keep the ball in his grasp as he looked to set Cuthbert free.
The Lions did get some reward for their heavy pressure in the shape of a second Farrell three-pointer when Adam Jones got the better of close friend Paul James at scrum time but they deserved even more when they again went close to breaching the Baa-Baas line after 23 minutes.
Farrell's cross kick was beautifully picked out of the air by Jamie Roberts, with the ball then recycled back to the left where Justin Tipuric's pass put O'Connell into space. The man who led The Lions in the absence of Sam Warburton attempted to find Hogg with a short offload but the Scottish full back knocked on before he could sprint clear.
O'Connell quickly went from creator to scorer, though, as he grabbed the first try from a close-range ruck. Phillips' power and pace was the catalyst as his sniping run was halted a centimetre or so shy of the line after an avalanche of phases, with O'Connell biding his time to avoid the attentions of Sergio Parisse and Samu Manoa in powering over the top of a mass of bodies.
Walsh asked the TMO for clarification but, when it came, the news was eventually what The Lions had been waiting for and Farrell duly added the extras to make it 13-3 after 27 minutes.
Having had to show patience before celebrating their first try, The Lions notched up their second with just three more minutes on the clock as Phillips sliced through yet again. The Lions' Test scrum-half from 2009 handed off Parisse after picking up possession 25 metres out and then successfully stretched for the line despite the best efforts of his Bayonne clubmate Joe Rokocoko, who did all he could to stop Phillips just short once more.
Farrell's second conversion left The Lions with a 20-3 advantage and the England fly-half extended that gap with a well-struck penalty from just inside opposition territory on the stroke of half time.
With the Baa-Baas having had their resistance severely tested and eventually broken in the latter stages of the first period, The Lions ran away with the contest in the second 40 minutes.
Gatland's troops crossed for a further six tries, with Phillips kicking things off with another fine individual effort just two minutes after the break. Phillips picked the perfect inside line after isolating Sam Jones when Lydiate claimed off-the-top lineout ball and, with Farrell converting from out wide, The Lions hit 30 to leave the Barbarians with a mountain to climb.
That hill threatened to get even higher six minutes later but Rokocoko held Tipuric up over the line after Mako Vunipola's offload looked set to provide another Lions score.
The Lions then produced a stunning counter attacking move that almost produced one of the tries of the season as they again made a mockery of the Australian media's description of the squad as 'slabs of red meat'. Hogg and Cuthbert exchanged passes on their own 22 to set things off, with the ball then spread wide left from where Maitland cleverly chipped into space. Toby Faletau picked up and offloaded to Farrell in one movement but the playmaker's overhead pass was then picked off by Payne just as Maitland looked destined to score.
Instead it was the Baa-Baas who were next to cross the whitewash as they breached The Lions' line for the first and only time through replacement scrum-half Kahn Fotuali'i. The Samoan finished off Rokocoko's incisive break and the gap was cut to 30-8.
But that minor frustration was just a blip as far as The Lions were concerned as they dominated the final 24 minutes to score a further five unanswered tries.
Jonathan Davies grabbed his first score in Lions colours after hacking on after Roberts put pressure on Nick Evans from an overthrown lineout on 57 minutes, before Cuthbert crossed for his brace on 60 and 68 minutes.
Both of the Welsh wing's scores came down the right-hand side, with Cuthbert himself starting the momentum for his first with a big hit on Parisse from which Roberts powered through to hand his team-mate a free run to the line.
Conor Murray then matched Phillips by breaking clear down the blindside before drawing the last defender and sending Cuthbert galloping clear from 25 metres out.
Jonathan Sexton missed both conversion attempts from Cuthbert's tries after succeeding with the extras from Davies' earlier effort so The Lions were still just short of a half century at 47-8 with 12 minutes left to play.
They passed that substantial milestone, though, when Lydiate was driven over from a short lineout with five minutes left and there was still time to put an even greater gloss on the scoreline as Jones dotted down in the final move of the match.
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