David Warner smashed his first international ton since completing a ball-tampering ban as Australia edged Pakistan by 41 runs at Taunton.
Warner (107) and Finch (82) got Australia to a cracking start with an opening stand of 146.
Mohammad Amir then pulled Pakistan back into the game with career-best five-wicket haul but Australia still got to 307, which ultimately proved too much for their opponents.
Pakistan suffered a middle-order collapse which saw them lose three wickets for 11 runs but Sarfaraz Ahmed (40) and Wahab Riaz (45) stitched a 64-run partnership to give them a ray of hope.
Australia vs Pakistan Highlights
But Mitchell Starc had Wahab caught behind got off a thin edge, and two balls later, Amir dragged the ball back onto the stumps.
Glenn Maxwell ended the proceedings with a stunning direct hit to get rid of Sarfaraz.
Pakistan's chase suffered a setback when Fakhar Zaman was caught off an upper cut to Kane Richardson at third man.
Babar Azam looked in sublime touch, with 28 of his 30 runs coming in boundaries, before he top-edged to deep backward square leg.
Finch, an occasional slow left-arm spinner, grabbed the all-important wicket Hafeez (46) off a rank full toss straight to the fielder at deep square leg.
Shoaib Malik was nipped out by Pat Cummins with a sharp in-dipper as Pakistan were reeling at 147-5 at one stage.
"I'm very disappointed," said Sarfaraz. "We lost three wickets in 15 balls and that's why we lost.
"We conceded too many runs in the first 20 overs apart from Mohammad Amir. We came back and restricted them well but it was a 270-280 pitch.
"We made some runs and got starts but we've got to convert them and go long. If you want to win matches your top four must score runs."
Australia have three wins in four games while Pakistan are struggling so far, with just one win in their four clashes.