The Suns made 75 percent of their shots in the third quarter, outscoring the Mavericks 32-20.
Dallas star Dirk Nowitzki missed his third straight preseason game with a swollen right knee.
The 11-time All-Star said Monday he will limit workouts to time in the pool and on the elliptical machine over the next few days. He wants to avoid surgery and hopes the reduced workload will bring down the swelling.
The Suns made 12 of 16 shots in the third to take an 81-74 lead after the quarter.
Dudley made a 3-pointer to tie the game at 61.
Dragic, who signed with Phoenix in the offseason and will replace Nash at point guard, hit a 12-footer to give the Suns the lead for good at 65-63 with 7:30 left in the third.
Dudley made a reverse layup to stretch Phoenix's lead to seven, and Dragic hit a 3-pointer to give the Suns an 81-71 advantage.
Mavericks reserve guard Delonte West played after being reinstated from a brief suspension.
After Monday's 123-104 win over Houston, Dallas coach Rick Carlisle suspended West without offering any details. The ban was lifted Tuesday night after the ninth-year guard met with Carlisle and president of basketball operations Donnie Nelson.
West entered the game midway through the first quarter and finished with seven points and four assists in 19 minutes. He also went into the stands early in the fourth quarter trying to save a loose ball.
Dahntay Jones and Mayo finished an 8-0 run with consecutive 3-pointers to give the Mavericks a 54-49 halftime lead.
Mavericks reserve guard Rodrigue Beaubois missed his third straight game with a left ankle sprain.