Barbosa made five of seven 3-pointers, and Phoenix newcomer Grant Hill added 24 points on the heels of his 21-point effort against New York two nights earlier.
Shawn Marion scored 11 of his 21 points in the final 6 1/2 minutes and Boris Diaw had seven of his 10 down the stretch in relief of Amare Stoudemire, who drew four fouls in a 3:18 minute span of the fourth quarter.
Steve Nash had 10 points and 15 assists for the Suns, and Stoudemire added 14 points.
Ben Gordon scored 24 and Luol Deng had 23 for the Bulls, who rallied from 14 down in the third quarter to take the lead in the fourth. Chicago fell to 1-6 to start the season.
Diaw's three-point play with 7:28 to go put Phoenix ahead for good, 88-85. The Bulls scored to get within one, then Marion's inside basket on a pass from Hill triggered a 9-0 run that put Phoenix in control for good.
The decisive spurt featured Barbosa's final 3-pointer, on a kickout pass from Nash on a fastbreak, to give the Suns a 93-87 lead with 6 minutes to play.
The Suns shot out to a 15-6 lead and were up by 15 in the second quarter.
But the Bulls finished the third with an 11-1 run to tie it at 75 on Kirk Hinrich's 16-footer with 21 seconds to go in the period. Chicago took a 76-75 lead when Tyrus Thomas made one of two free throws to start the fourth.
There were four lead changes and five ties before Diaw's three-point play.
Barbosa, who scored 16 in the second quarter, hit three straight 3s in an 11-3 spurt that put the Suns ahead 56-41 with 1:09 left. But Gordon sank a 3-pointer and Deng added a layup to cut it to 56-46 at halftime.
Hill, who shot 7-of-8 in Phoenix's victory over New York on Tuesday, had 17 in the first half against Chicago, 11 in the second quarter.
Five consecutive points by Marion, on a 3-pointer and a driving runner, gave Phoenix a 62-48 lead before the Bulls launched their rally.
Signed as a free agent in the offseason, Hill has quickly acclimated to the Suns' ultra-uptempo style. He had a fastbreak layup on a half-court lob pass from Barbosa, the last of his six straight points that put Phoenix ahead 36-32.
Raja Bell was back in the Suns lineup after missing two games because of a sprained right ankle. ... Marion arrived later than usual because he was involved in a car accident on his way to the arena. ... Stoudemire turns 25 on Friday. ... Chicago assigned rookie guard JamesOn Curry to Iowa of the NBA Development League.