When Finch holed out Australia needed just four runs and despite Sean Abbott fell for nought in the same Shaheen Shah Afridi over, Ben McDermott hit the winning boundary in his 19-ball 22 not out to achieve Australia's fourth successive win in T20Is against Pakistan.
Finch, who hit six boundaries, put on a brisk 40 for the opening stand with Travis Head (14-ball 26 with three fours and a six) and another 44 for the second with Josh Inglis who made 15-ball 24 with two fours and a six.
Pakistan did pull back with leg-spinner Usman Qadir (2-33) and pacer Mohammad Wasim (2-30) brought the home team back in the game.
But Marcus Stoinis (nine-ball 23 with five boundaries) and McDermott ensured they supplement their skipper to achieve the chase.
"I thought the way our bowlers attacked that game were fantastic," said Finch. "Nathan Ellis had a tough start but showed real character and we took wickets upfront."
Pakistan skipper Azam rued loss of wickets after a good start.
"We had a very good start but then lost couple of wickets that took away the momentum," said Azam. "We lost momentum and in the end were 20-30 runs short."
Pakistan made a brisk start.