After opting to bat, Conway (101) and Williamson (85) gave New Zealand a solid start with a 181-run second-wicket stand but they failed to capitalise fully as Mohammad Nawaz took 4-38 and the touring side were bowled out for 261.
Pakistan lost two early wickets before captain Babar Azam steadied the ship with an innings of 79.
He lacked support from his teammates, however, and Tim Southee and Ish Sodhi took two wickets each as New Zealand wrapped up victory with seven overs to spare.