Kevin Durant scored 35 for the Thunder, who trailed by 28 in the first half but cut the lead to six early in the fourth quarter before Barbosa and the Suns blew the game open for good.
Barbosa moved into the starting lineup after Stoudemire underwent surgery Friday to repair a partially torn retina in his right eye.
The speedy Suns guard also had seven assists and tied his career best with six steals.
Phoenix improved to 3-0 since Terry Porter was fired and replaced by Alvin Gentry, who reinstated the Suns' super-speed style that the team played so well under Mike D'Antoni.
Since that change, the Suns beat the Los Angeles Clippers 140-110 on Wednesday night and 142-119 on Thursday. Those two wins came with Stoudemire, who won't be able to resume physical activity for about eight weeks, meaning he could well be out for the rest of the season.
Gentry decided to go small with the All-Star power forward out, inserting Barbosa at guard, shifting Richardson to small forward and Grant Hill as the starting power forward.
Barbosa made 16 of 21 shots, 5-of-6 in the fourth quarter. He was 5-of-7 on 3-pointers. Matt Barnes had 14 points and career-best nine assists for the Suns.
The competition gets a bit tougher when Boston comes to Phoenix on Sunday.
Down 97-76 after Barnes' inside basket with 3:40 to go in the third quarter, the Thunder climbed back in it with a 15-2 run that cut the lead to 99-91 on Earl Watson's 3-pointer with 1:16 left in the period. The Suns led 103-93 entering the fourth.
Nenad Kristic made a pair of 20-footers in a 6-0 spurt that cut the lead to 105-99 after Nick Collison's inside basket with 9:40 to go.
Richardson's fourth 3-pointer - in five tries - capped a 12-2 run that put Phoenix ahead 122-105 with 4:26 to play, and Oklahoma City was finished.
