Under pressure after experiencing a lean run of form, the Australian captain hit 55 as the tourists passed Pakistan's total of 8-162 with five balls to spare at a sold-out Gaddafi Stadium in Lahore.
The 16th half-century of Finch's T20I career was his best score in the format since scoring an unbeaten 79 in March 2021 against New Zealand.
Finch was caught on the boundary with Australia needing just five more runs, before Ben McDermott hit the winning shot at the start of the last over.
"I always knew that I could play a little bit so I think the most vulnerable you are in the game as a batsman is your first handful of balls," Finch said.
"Just in the last few innings I didn't get through that initial period.
"I always had confidence, I always had faith and it's one of those things you go through some lean patches at times.
"Everyone can seem to panic at times, especially as players get a bit older, people panic quicker.
"The support I've had from the players and the coaching staff has been fantastic.
"Against the new ball there wasn't quite as much swing there so there was a slight tinkering with technique but nothing uncommon."
Finch was the mainstay of Australia's innings, but Travis Head (26 off 14 balls) and Marcus Stoinis (23 off nine) both chipped in with timely and vital cameos.
He will now head to the Indian Premier League to play with his ninth franchise after Kolkata Knight Riders signed him as a replacement for England's Alex Hales.
After winning a hard-fought Test series 1-0, Australia stumbled in the ODIs, but their under-strength line-up was too strong for Pakistan in the one-off T20.
Finch sent Pakistan in after winning the toss and openers Mohammad Rizwan and Babar Azam raced to 0-56 at the end of the first powerplay.
But Cameron Green, on T20I debut, helped stop Pakistan from posting a near-unbeatable score by dismissing Rizwan and Fakhar Zaman in consecutive balls in the eighth over.
Despite the best efforts of captain Babar Azam (66), Pakistan could never regain their early momentum, when they appeared to be on record pace.
Tasmanian paceman Ellis recovered from an early battering to cause all sorts of problems for Pakistan's middle-order to finish with career-best figures of 4-28.
Ellis was on a hat-trick, but came up short as Australia burned a review on challenging a not-out decision for a caught-behind.
Australia handed T20I debutants to Green, Marnus Labuschagne and Ben Dwarshuis as they continue their preparation for the World Cup defence on home soil in October.
Green showed he could be a valuable back-up for last year's World Cup hero Mitch Marsh, but Labuschagne (2) and Dwarshuis (0-42) had forgettable first outings.
After leaving COVID-19 isolation on Monday, Josh Inglis hit a promising 24 after slotting back into the Australian top-order after a breakout T20 series against Sri Lanka in February.
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