Pakistan wasted a strong start from openers Sarfraz Ahmed (65) and Ahmed Shehzad (61) as they were bowled out for 215 in the final over after captain Misbah-ul-Haq had won the toss and opted to bat.
Australia lost a couple of early wickets in their run-chase but a fourth-wicket partnership of 85 between Maxwell and captain George Bailey (28) proved the key.
Bailey was run out before Maxwell, who was dropped on two by Umar Akmal at slip off debutant Raza Hasan, top-edged a sweep off Zulfiqar Babar for Asad Shafiq to complete a running catch.
All-rounder James Faulkner (26 not out) and wicketkeeper Brad Haddin (17 not out) then struck a few lusty blows to take Australia past their target with 6.4 overs to spare.
“I was just playing my role,” said man of the match Maxwell at the presentation ceremony after hitting nine fours and a six in an 81-ball knock.
“Today I came in at a different situation with the ball spinning … I just tried to play to the situation and get the team close to the total.”
Earlier, Pakistan reached 126 without loss in 25 overs but the innings then imploded as they lost 10 wickets for 89 runs.
The openers were removed in successive overs before Misbah’s untimely run out for 15 robbed the Pakistan innings of momentum on the spinning track.
The Australian bowlers, led by Mitchell Johnson’s three for 40, turned the screws and made scoring difficult for the batsmen on the slow pitch.
“Poor display in the last 25 overs,” Misbah said. “If we had batted normally we would have scored 250 … we could have defended that.” -Reuters