Phoenix was missing two-time MVP Steve Nash, who sat out with a shoulder injury that kept him out of the second half in Monday's 113-108 win over Denver.
Barbosa stepped in at point guard, tallying seven assists and six rebounds, and hitting key shots in the second half.
"Barbosa really came out in the second half and picked it up," Stoudemire said. "He's been there all season long for us."
Stoudemire scored the first five points of the extra period and Barbosa knocked in back-to-back 3-pointers as the Suns held off the Blazers.
Portland missed a very good opportunity to win the game in regulation. With the score tied at 96, Travis Outlaw drove to the hoop but missed a layup with 4 seconds left.
"I told Travis after the game you've got to power that shot to the basket," coach Nate McMillan said. "I couldn't believe it when I saw him get a path to the basket but it didn't go in. You can't finesse a shot like that."
Stoudemire, coming off a 36-point, 13-rebound game against Denver, also had nine rebounds, just missing his fourth straight double-double.
The Suns looked at times like they might run away with the game. They led by as many as 13 in the first quarter and had a 40-26 lead in the second, but Portland closed the gap, getting within 48-45 after Jack converted a three-point play.
Phoenix led by four at the half and started fast in the third, leading 73-61, but the Blazers went on a 14-4 run to get within two.
"We just didn't have constant knowledge of what we were doing without Steve [Nash] in there," Phoenix coach Mike D'Antoni said.
Randolph, who made his first eight shots, cut the lead to 85-84 with a three-point play at 5:52 in the fourth, and Martell Webster tied the game at 87 with a 3-pointer.
Barbosa stole the inbound pass and then scored on a rebound put-back of a Stoudemire miss. The next time down the floor he swished a 3 to extend the lead to five points.
Randolph was fouled following a Roy miss and converted two free throws with 57.3 seconds left to tie the game at 93.
Barbosa knocked in a 3 from the corner with 38.9 seconds left to give the Suns a 96-93 lead, but Udoka took a pass from Roy and hit a wide-open 3 for the Blazers.
Nash said he first felt soreness in the shoulder when he got out
of bed on Thursday and it has worsened daily since.
After Tuesday's game, the Suns don't play again until Friday,
when they are home against Atlanta.
Nash is averaging career highs of 19.3 points and an NBA-leading
11.8 assists per game.
Portland made 27 of 30 free throws. ...The Blazers wore a black band on their jersey for former team executive Stu Inman, who died last week at 80. ... Blazers starting center Joel Przybilla played six minutes and didn't score. Przybilla, who signed a 5-year, $32 million deal before the season, is averaging just 2.1 points and 4.0 rebounds this season. ... Phoenix has won six straight against the Blazers for the first time since Nov. 15, 1994 to Nov. 14, 1995.