England, set a modest 201 for victory, finished on 201 for two with 13 balls as they won the series, going 3-0 up in this five-match campaign after Wednesday's clash in Birmingham was washed out.
Jonathan Trott (64 not out) and Ravi Bopara (33 not out) shared an unbroken stand of 65 as England recorded their ninth successive one-day international victory with 13 balls to spare.
Ian Bell (69) and Warwickshire colleague Trott had previously put on 66 for the second wicket.
Bell's 94-ball knock, featuring nine fours, meant that in five innings since taking over as an opener following Kevin Pietersen's limited-overs retirement he had scored 364 runs at an average of 72.8 with one hundred and three fifties.
Meanwhile Trott's effort, in his 47th innings, saw him become the fourth-fastest batsman to 2,000 ODI runs behind South Africa's Hashim Amla (40 innings) and Pakistan's Zaheer Abbas and Pietersen (both 45).
Earlier, fast bowler Finn's haul of four for 37 in his maximum 10 overs sparked an Australia top-order collapse as the visitors struggled in overcast conditions after losing the toss.
That Australia, still the world No 1 side in this format, recovered from a lowly 96 for six to 200 for nine was largely down to David Hussey's 70.
Alastair Cook, the England captain, and Bell made a sound start to the chase.
Bell later eased fast bowler James Pattinson, recalled alongside fellow seamer Ben Hilfenhaus in place of left-arm quick Mitchell Johnson and spinner Xavier Doherty, to the cover boundary with typical elegance.
Australia's problems intensified as all-rounder Shane Watson and Lee both went off injured, meaning captain Michael Clarke had to use part-time bowlers.
But they did reduce England to 70 for one when Cook (29) drove McKay to substitute Steven Smith at short extra-cover.
Bell had looked in command until he chopped on to seamer McKay, the only Australia bowler to take wickets in this match with two for 29. England would have had even fewer runs to chase had they not dropped four catches, with Hussey missed on 15 and 29.
Finn, twice on a hat-trick, ended just shy of his career-best four for 34 which he achieved twice against Pakistan in the UAE in February. England saw Finn claim two wickets in successive balls as Australia slumped to six for two in the sixth over. It was Finn's third four-wicket haul in ODIs.
The series concludes with a day/night match at Old Trafford on Tuesday.
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