Al Jefferson scored 32 and C.J. Miles 19 for the Jazz, who led by eight in the fourth quarter before an 11-0 run put the Suns ahead for good.
Steve Nash had 20 points and 14 assists for Phoenix, which has won seven of its last nine. The Suns playmaker passed Gary Payton for seventh on the NBA's career assists list. With three 3-pointers, Nash also tied Eddie Jones for 10th on the career list.
The Jazz, with four players out with injuries, has lost 12 of 16 and fell to 0-2 since Sloan resigned and was replaced by Tyrone Corbin, both losses to the Suns.
For the second game in a row, the Suns led entering the fourth quarter, only to have the visitors take command with a big run. Unlike Sunday's loss to Sacramento, Phoenix came back this time.
Down 76-73, Utah opened the fourth quarter with a 16-5 run to go up 89-81 when Jefferson made one of two free throws with 7:12 to play. But Phoenix scored the next 11, including a pair of 3-pointers by Mickael Pietrus and one by Frye, to regain the lead 91-89 with 3:35 remaining.
Deron Williams, who made just 2 of 11 shots, sank a 3-pointer to cut the lead to 99-98, then Frye missed one of two free throws with 9.9 seconds to play to make it 100-98. After a timeout, Paul Millsap found himself trapped inside and lost the ball to Nash, who was fouled.
Nash's two free throws put Phoenix ahead 102-98 with 2.9 seconds left. Miles' 3-pointer at the buzzer made it a one-point finish.
Frye scored 19 in the first quarter on 7 of 7 shooting, five of them 3-pointers.
He made his first eight shots before missing a 3 near the end of the first half. Still, the Suns trailed 53-50 at the break.
Utah's 9-2 spurt at the start of the second quarter put the Jazz up 36-31. After Nash committed the 13th of Phoenix's 14 first-half turnovers, Miles' layup put Utah ahead 47-39.
Frye's 22-footer capped an 8-0 run that put Phoenix ahead 70-63 and the Suns led 76-73 entering the fourth quarter.
Utah was without Andrei Kirilenko (sprained right big toe), Raja Bell (strained left calf), Ronnie Price (sprained right big toe) and Mehmet Okur (strained lower back).
Notes: The Suns' Goran Dragic returned after missing seven games with a cut on his left foot. ... Frye's previous career best was 30, set three times with the New York Knicks. ... Corbin was a finalist for the Suns coaching job that went to Terry Porter in 2008. ... Phoenix has won 12 of 17.