Reserves Jeff Green and Barbosa each scored 16 points for the Celtics, who won for the fifth time in six games. Kevin Garnett added 11 points and eight rebounds.
Rondo left the game for good in the third quarter with the ankle injury after getting 10 assists - reaching double-digits for the 32nd straight game.
With Rondo sitting on the bench for the final quarter, Green and Barbosa each played the entire 12 minutes, with Green scoring seven points.
"Barbosa was terrific," Boston coach Doc Rivers said. "I mean, he bailed us out."
Rivers said he "had no idea" if Rondo would play Thursday in Brooklyn.
When Avery Bradley returns from offseason shoulder surgeries, Barbosa will be in a mix of five guards along with Rondo, Jason Terry and Courtney Lee.
Paul Millsap led the Jazz with 20 points and 12 boards. Al Jefferson had 13 points and 14 rebounds.
"Tonight we proved that we can fight. That's the difference between the Celtics and us right now. They know how to seal the game," Jefferson said. "That's what we've got to get to. Playing these type of teams, it's going to help us get there."
Mo Williams missed three shots in the closing minute for Utah, which was coming off a triple-overtime win at Toronto on Monday but fell to 1-5 on the road.
In other games *
At Houston, James Harden scored 30 points, Omer Asik added 15 points and 12 rebounds, and the Rockets held off the New Orleans Hornets, 100-96. Greivis Vasquez scored a career-high 24 for the Hornets, despite hurting his left ankle in the third quarter and briefly leaving the game. Anthony Davis had only eight points on 2-for-7 shooting.
* At Minneapolis, Kemba Walker's jumper with 0.7 seconds left lifted the Charlotte Bobcats to an 89-87 victory over the Minnesota Timberwolves. Walker finished with 22 points, five assists and four rebounds, and his 19-footer over the outstretched arm of Alexey Shved helped the Bobcats to their first three-game winning streak in nearly 20 months.
* At Oklahoma City, Rudy Gay scored a season-high 28 points, Zach Randolph added 20 points and 11 rebounds before getting ejected along with Kendrick Perkins during a late-game altercation and the Memphis Grizzlies beat the Thunder, 107-97. Marc Gasol chipped in 14 points as the Grizzlies continued the best start in franchise history by winning their sixth straight game.
* At Milwaukee, Brandon Jennings and Monta Ellis scored 16 points each to help the Bucks cruise to a 99-85 victory over the struggling Indiana Pacers. Beno Udrih had nine points, 10 assists and six rebounds for the Bucks.
* At Dallas, O.J. Mayo scored 25 points and Chris Kaman added 23 as the Mavericks held off Washington, 107-101. Jordan Crawford scored 21 off the bench for the winless Wizards.
* At Phoenix, Carlos Boozer had 28 points and 14 rebounds and Luol Deng and Joakim Noah added 21 points each to lead the Chicago Bulls to a 112-106 overtime win over the Suns, who got 24 points and 14 rebounds from Luis Scola.
The NBA players' association is continuing its appeal on behalf of Sacramento Kings center DeMarcus Cousins even after he has served his suspension.
The league suspended Cousins two games for confronting San Antonio announcer Sean Elliott in a "hostile manner." The exchange followed the Spurs' 97-86 victory in Sacramento on Friday night. Cousins is hoping to recoup his pay for each game missed. He lost about $70,500 of his $3.9 million annual salary.