Cook made 137 to lead England to a competitive total of 260 for seven in the day-night match and fast bowler Finn struck four times in a rapid opening spell as Pakistan crumbled to 130 all out in 35 overs.
It was an excellent start to the four-match series for England, who lost all three tests to Pakistan and were beaten 5-0 in their last one-day series in India.
Cook won the toss and batted confidently from the start, striking 14 fours in his third one-day international century.
The left-hander shared an opening partnership of 57 with Kevin Pietersen to give England a solid foundation before Pietersen fell for 14 to Shahid Afridi and Jonathan Trott was out first ball to the same bowler.
Cook and Ravi Bopara smoothly added 131 for the third wicket but Saeed Ajmal returned to the attack to have Bopara stumped for 50 and the off spinner also bowled Cook on the way to figures of five for 43 as England faltered in the closing overs.
Finn struck in his second over when he removed Mohammad Hafeez and Asad Shafiq with successive deliveries, and the rangy pace bowler also had Imran Farhat and Younus Khan caught by wicketkeeper Craig Kieswetter in an impressive spell.
Spinner Samit Patel trapped Pakistan captain Misbah-ul-Haq lbw for 14 and only a typically belligerent knock by Afridi, who struck 28 from 22 balls, lifted Pakistan past 100.
Finn finished with four for 34, Patel took three for 26 and Graeme Swann chipped in with two as England completed a morale-boosting win following their dismal performance in the tests.
The second match of the series is in Abu Dhabi on Wednesday.
