New Zealand posted 288-7 with Mitchell smashing 113 of 115 balls and forging a century-plus stand with Will Young, who scored 86.
Naseem Shah was the pick of Pakistan's bowlers, conceding only 29 runs in 10 overs and claiming two wickets with the last two balls of the New Zealand innings.
Pakistan's Fakhar and Imam-ul-Haq, who finished on 117 and 60 respectively, combined in a strong opening stand of 124 and skipper Babar Azam contributed 49.
Mohammad Rizwan remained not out on 42 as Pakistan reached the target with nine balls to go.
"The plan was to attack and stay long," said Fakhar, who was named Player of the Match. "If you survive the new ball, you must stay (till the end)."
The idea was to have at least one of the top three batters to bat deep, he said.
"Today was my day and hopefully, Imam and Babar will also play big knocks."
Mitchell said New Zealand managed a "par score" batting first on a low wicket.
"Pakistan have got some world-class bowlers and they showed that at the end there with their death bowling," he told reporters.
"Fakhar was dangerous, he took the game on at times, took some risks and he obviously got the rewards for that."
Rawalpindi also hosts the second ODI of the five-match series on Saturday.