Man-of-the-match Finch's stunning 156, the highest individual score in all Twenty20 internationals, was the cornerstone of Australia's 248 for six, itself the second-highest team score at this level.
England, who wrapped up a 3-0 Ashes series win with a draw at The Oval on Sunday, unsurprisingly lost early wickets when faced with such a colossal chase and finished on 209 for six.
Victory saw Australia go 1-0 up in a two-match series that concludes in Durham on Saturday.
Australia hadn't won a major international match since a 17-run victory over the West Indies at Melbourne on February 10 saw them complete a 5-0 one-day international series clean sweep.
Finch's runs came from just 63 balls and the 26-year-old Victorian opener, playing just his seventh match at this level, comfortably surpassed the previous Twenty20 world record of 123 set by New Zealand's Brendon McCullum against Bangladesh at Pallekele in September last year.
Remarkably, Finch got off the mark, reached his 50, hundred and 150 with sixes.
Indeed his 14 sixes were also a world record for the most in any individual T20 international innings, with Finch also striking 11 fours.
Finch's century was only the 10th in all T20 internationals, the first by an Australian and the first by any batsman at this level in England.
His extraordinary innings ended when he was bowled by seamer Jade Dernach who, in the midst of the mayhem, took three wickets for 34 runs in four overs.
In view of the run-spree to come, England, who won the toss, started well when dangerman David Warner lost control of his bat as he holed out off home Twenty20 captain Stuart Broad for one.
From then on, the innings belonged to Finch who struck an extraordinary six over cover on one knee off medium-pacer Ravi Bopara to the astonishment of a 20,000 capacity crowd.
His 50 came off 27 balls, with a straight six off Bopara, and he appeared on course to beat South African Richard Levi's record 45-ball century.
Finch had to be content with a 47-ball hundred, going to the landmark with a top-edged six, while McCullum's previous record was surpassed with yet another six, this time over cover off Broad.
Dernbach finally bowled Finch and removed Shane Watson as Australia fell short of Sri Lanka's T20 international record team total of 260 for six against Kenya in Johannesburg in 2007.
But Australia's total was the highest by one Test nation against another in this format of the game.
Australia opening bowlers Mitchell Johnson and Josh Hazlewood took four wickets between them as England slumped to 42 for four, with their side showing just two survivors from The Oval in Broad and Joe Root.
However, it was Root's commendable 90 not out -- his first T20 international innings -- off just 49 balls, including 13 fours and a six, that prevented a humiliating loss.
Australia gave an international debut to Pakistan-born leg-spinner Fawad Ahmed, a 31-year-old former refugee who only became an Australian citizen last month.
But the Victoria bowler went wicketless in a return of nought for 43 in his maximum four overs.