The victory pulled a youthful England ODI side level at 2-2 in a five-match series ahead of Sunday's finale at Bristol.
Livingstone's whirlwind 27-ball innings took England to 312-5 in a match reduced by rain to 39 overs per side after captain Harry Brook just fell short of successive hundred with 87.
Australia made a sound start to their daunting chase but lost six wickets for 28 runs in slumping from 68-0 to 96-6.
They were eventually dismissed for 126 inside 25 overs, with paceman Matthew Potts taking an ODI-best 4-38.
Fit-again opener Travis Head's 34 was the top score in an Australia innings where only four batsmen made it into double figures.
This was England's second-biggest win at this level over arch-rivals Australia in terms of runs, following their mammoth 242-run success at Trent Bridge in 2018.
"We've taken all the positives and the momentum and brought them here," player-of-the-match Brook, 25, told Sky Sports. "Livingstone, we know what he's capable of and he showed that tonight."
"We just identify the times we need to apply pressure."
'Gem of an innings'
Australia captain Mitchell Marsh accepted his side had been outclassed.
"We were a bit off today but fair play to England they put us under pressure," he said. "They out-played us today... Liam Livingstone played a gem of an innings."
England were 16 runs shy of 300 when Mitchell Starc began to bowl the last over of their innings.
But Livingstone, renowned as a big-hitter, then hammered the left-arm fast quick for 28 runs -- the most expensive over by an Australian bowler in men's ODIs -- including four sixes in five balls, before finishing with a four.
Livingstone started the 39th over by driving Starc into the pavilion for six.
And two balls later, Livingstone lofted Starc for six over midwicket and next ball drove him for another six. The fifth ball also went for six, via a monster pull, before Livingstone scythed the last ball of the over for four.
Livingstone struck three fours and seven sixes in total as Starc finished with hugely expensive figures of 0-70 in eight overs.
Australia, who saw their run of 14 straight ODI wins ended by England at Chester-le-Street on Tuesday, were up against it after that late blitz.
But Head, who made a superb 154 not out in Australia's series-opening win in Nottingham, launched Brydon Carse for a huge six onto the roof of the Mound Stand.
Carse, however, had his revenge when he bowled left-hander Head for 34 from around the wicket.
Australia's 68-1 then became 75-2 when Potts had star batsman Steven Smith caught behind for just five.
After Marsh was bowled for 28 by a superb delivery from express fast bowler Jofra Archer that nipped past his outside edge, Carse struck twice in the 15th over to remove Josh Inglis and Marnus Labuschagne.
Potts just missed out on a hat-trick before leg-spinner Adil Rashid ended the match by bowling last man Josh Hazlewood for a duck.
Earlier, Brook -- fresh from a match-winning maiden ODI century at Chester-le-Street -- came in with England faltering at 71-2.
But he responded with an exhilarating 58-ball innings including 11 fours and a six.
Australia thought Brook had been caught behind down the legside for 17 by Inglis, playing his first match of the series after injury, off Starc.
But an umpire review made it clear the ball had bounced into Inglis's gloves -- with boos ringing round Lord's as the replay was shown on the giant screen.
It was the second controversial incident involving an Australia wicketkeeper at Lord's in as many matches after Alex Carey -- playing as a batsman only on Friday -- stumped Jonny Bairstow during an Ashes Test last year when the England batsman believed the ball was dead.
Brook fell when he holed out in the deep off leg-spinner Adam Zampa.
