The Suns, who didn't make a 3-pointer in their opening loss to Atlanta on Wednesday, went 9-of-21 from beyond the arc against Utah (1-1) and didn't trail after the opening minutes of the first quarter.
Raja Bell scored 11 points for Phoenix (1-1).
Kyle Korver scored 16 points and Mehmet Okur had 12 points and eight rebounds for Utah, which opened with a victory over the Los Angeles Lakers on Tuesday.
Deron Williams, a U.S. Olympian this summer, had nine points and five assists as he played the entire first and third quarters.
Boris Diaw added nine points and eight rebounds for Phoenix, which outrebounded Utah 45-35, the second straight time Utah was outrebounded by double figures.
Phoenix led by as many as nine points in the third quarter and pushed the lead to 14 points late in the fourth quarter, largely because of Tucker. The second-year forward from Wisconsin scored 11 points in the final quarter.
Barnes, an offseason free agent signing, scored eight points in the final quarter for Phoenix, which shot 48.6 percent from the field.
Guard Morris Almond scored 10 points and forward Paul Millsap had a team-best 11 rebounds for Utah.
Both teams struggled from the free throw line, making just 61 percent. Phoenix's Shaquille O'Neal missed his first three free throws and finished with five points.
The Jazz played without All-Star forward Carlos Boozer, who missed his second consecutive game resting a sore left hamstring. The Suns were without Amare Stoudemire (torn iris in right eye) and Leandro Barbosa (family reasons) for the second straight game.