Pakistan, who were whitewashed in the preceding test series and lost the first ODI in Brisbane, reached their target with 14 balls to spare after their bowlers had bundled out the hosts for 220 in the penultimate over of their innings.
It was the first win for Pakistan against Australia in Australia across all formats since their victory in an ODI at the WACA in January, 2005.
Mohammad Amir was the most successful Pakistan bowler with three wickets for 47 after left-arm fast bowling colleague Junaid Khan, playing his first ODI in over 19 months, put Australia on the backfoot by sending back the openers cheaply.
Pakistan's spinners, led by left-armer Imad Wasim (2-37), then bowled tightly on a sluggish Melbourne Cricket Ground pitch to keep Australia shackled.
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Australia captain Steve Smith top-scored for his side with 60 but a number of other batsmen got out after promising starts.
Hafeez, who was not part of the original ODI squad and arrived in Australia as a late addition, was handed the captaincy after regular skipper Azhar Ali was ruled out with a hamstring injury.
The right-handed opener struggled against Australia's pacemen initially and Smith dropped him on nought in the slips off Mitchell Starc.
Hafeez, who survived again on 55, batted fluently after his initial troubles and set up the chase with an opening stand of 68 with Sharjeel Khan (29) before a second-wicket partnership of 72 with Babar Azam (34).
Experienced Shoaib Malik, who was given out caught behind off Pat Cummins on 10 but successfully overturned the decision on review, took over the responsibility of getting Pakistan home after they lost Hafeez and Azam in successive overs.
Malik remained unbeaten on 42 after adding a crucial 53 for the fourth wicket with Asad Shafiq (13).
The third ODI of the series will be played in Perth on Thursday.
(Reporting by Sudipto Ganguly in Mumbai; editing by Toby Davis)
