The 27-year-old was joint overnight leader with Fisher in Doha but by the turn on the final round, Pepperell had established a three-shot lead and looked in complete control.
He maintained a two-shot advantage until the 17th, but a Fisher birdie on the par three set up a nerve-wracking finish.
Fisher even had a chance on the final green to force a play-off but missed his birdie putt, leaving Pepperell to win with a par on the 18th.
He threw his ball and cap away in relief as he finished with a two under par 70, his worst score of the week, taking the title on 18 under.
Fisher finished second, one shot behind after shooting a 71. It was his first top three finish since 2014.
Sweden's Marcus Kinhalt finished third on 16 under par.