Still coach Jerry Sloan kept urging his team to "keep on fighting, keep on fighting," Utah forward C.J. Miles said.
Fight they did, erasing an 11-point deficit with 7:43 remaining to beat the Suns 116-108 on Thursday night.
Williams scored 13 of his 27 points during a 41-point fourth quarter and Okur finished with 24 points.
The Jazz trailed 96-85 early in the fourth quarter, then reeled off a 13-2 run, capped by a 3-pointer by Okur, to tie the game.
They took the lead for good on a 3-pointer by Williams with 1:36 remaining that made it 109-106.
Overall, Utah finished 12 of 24 from 3-point range, including 7-of-11 in the fourth period.
"Basketball is a game of runs," Miles said, "and we got ours at the right time."
The Suns, meanwhile were only 5 of 19 from behind the arc.
"I think we did a good job of not letting them hit the home run most of the night," Sloan said.
Carlos Boozer had 15 points and 15 rebounds for Utah, and Miles scored 15.
"It seemed like they [the Suns] had the momentum in the first three quarters," Boozer said. "We've played this team twice and both times they were able to jump out and somehow we were able to come back and beat them.
"But we can't keep up that trend."
The victory kept the Jazz in fourth place in the Western Conference, 1 1/2 games ahead of the Suns.
"You never know what's going to happen and who's going to end up where in the playoffs," Boozer said. "Especially when you're playing in the 'Wild West' where every game is monumental."
"To come back the way we did says a lot about the character of our team," Williams said.
Phoenix coach Alvin Gentry was mostly satisfied with his team's play.
"I thought we played well for about 45 minutes, but the last three minutes is where it fell apart," he said. "We just couldn't finish the job off. You have to give them credit. They're a tough team and they did a good job of executing down the stretch."
"I don't know if fatigue set in," Stoudemire said, "but we played well most of the night. We just didn't play well defensively down the stretch. We just have to key in defensively in the fourth quarter to get some stops."
The Suns took control in the first half with a blistering shooting performance (13-for-19, 73.7 percent) and an 11-0 run in the opening quarter. After Okur hit the first field goal, the Suns ran off their burst, with Robin Lopez making three shots.
The Jazz never got closer than five points after that in the half and needed a 3-pointer by Sundiata Gaines at the buzzer to trail 57-49 at intermission.
Nash, who scored only two points against the Los Angeles Clippers on Wednesday night, already had 14 at the half, along with eight assists, directing the Suns' offense with his usual efficiency.
At the break, the Suns still were shooting a dazzling 64.9 percent (24-for-37). What kept Utah reasonably close was its offensive rebounding. Nearly half the Jazz's total 23 rebounds (11) were off the offensive glass and they scored 11 second-chance points.
Phoenix's biggest lead in the first half was 13 points, 37-24, on a three-point play by Jared Dudley with 1:30 gone in the second quarter.
Utah was hurt in the half when Williams and Andrei Kirilenko each incurred three personal fouls.
Stoudemire really asserted himself in the third quarter, scoring 15 points on a variety of shots -- including three dunks, a jumper, a drive, a layup and three foul shots, two of them completing three-point plays.
Okur kept the Jazz reasonably close -- they trailed 86-75 at the end of the period -- with 10 points. The Suns' turnovers -- they had a total of 15, leading to 16 points, also helped the Jazz.
But it was the Suns' extraordinary shooting -- they were 34-for-53, 64.2 percent, at that point, that kept them safely ahead.
Suns guard Goran Dragic missed his first game of the season because of an injured right ankle, sustained during Wednesday night's game at the Los Angeles Clippers. ... This was the start of a seven-game, 17-day homestand for the Suns, their longest since 1993. During that stretch, the Suns will face Utah twice, and Indiana, the Los Angeles Lakers, Charlotte, Minnesota and Portland once each.