The two-time MVP sat out with back spasms, but Barbosa filled in admirably. He connected on 10 of 18 shots and scored 22 points in the second half.
Phoenix also overcame the loss of big man Amare Stoudemire, ejected in the second quarter after his second technical.
No problem, O'Neal said.
"All you've got to do is throw it in to the most consistent big man in the history of the game," he said with a smile. "I'm proven."
O'Neal had 13 rebounds and three blocks, and his fourth-quarter dunk moved him past Oscar Robertson on the career scoring list with 26,711 points. O'Neal was 8-of-13 from the field and missed only two of his 10 free throws.
"I can remember growing up, my father bragging about Big Oscar," O'Neal said. "For me, it's a credit to my parents for raising me well, and a credit to me for just working hard and playing hard.
"I'm really upset with myself. I've missed about 5,000 free throws. If I had only made half of those, I'd probably be No. 3 right now."
Hakim Warrick led Memphis with 25 points, while Marc Gasol finished with 16 points and 13 rebounds. Rudy Gay had 14 points and O.J. Mayo scored 12, but he missed 16 of 20 shots -- including his last 13.
"We didn't play too well in the fourth quarter. Maybe we lost focus a little bit," Mayo said.
Memphis fell apart down the stretch, hitting only three of 18 shots in the fourth quarter, including misfiring on its final seven from the field.
"Shaq controlled the paint, and we did a good job on their main two: Mayo and Gay," Hill said. "We did a great job. We've shown that we can do it.
"It's good to know that when you need to play defense and you need to get stops, you're capable."
The Grizzlies held an eight-point lead in the fourth, but that was erased by nine unanswered Phoenix points, eight of them by Barbosa on a pair of 3-pointers and a layup for the Suns' first lead since the early stages. From there, Memphis' inaccuracy from the field allowed the Suns to extend the lead to double digits.
"Obviously, Shaq made a presence down there," Gay said. "Barbosa came in and knocked down a couple of 3s and before you know it, we're down."
O'Neal was perfect on his first four shots from the field and his first four free throws. That kept the Suns within striking distance as Memphis led by nine in the first half.
The Suns had the bulk to overpower Memphis inside with O'Neal and Stoudemire. Memphis was limited because Darko Milicic was out with a broken right hand, and in the first quarter rookie Darrell Arthur took an elbow to the face, sending him to the bench for the night with a concussion. Memphis even used Hamed Haddadi, the 7-foot-2 rookie from Iran who saw his first action of the season.
The Suns' inside dominance was evident with a 36-16 advantage in the paint in the first half, including 16 points from O'Neal. Phoenix ended the night with a 58-18 edge in the paint.
But the Suns' size advantage was offset somewhat in the second when Stoudemire was ejected with two technicals.
Stoudemire disagreed with the call that gave him his third foul. He expressed his displeasure to official Bill Kennedy, drawing his first technical. Stoudemire then added more colorful language, drawing another technical from Kennedy and sending Stoudemire to the locker room for the night.
He left with only eight points, the first time this season Stoudemire has not reached double figures.
"We really needed [Shaq] without Steve and Amare," Suns coach Terry Porter said. "He was the one guy that we could go to knowing he would make plays for us. He took the challenge."
Memphis held a 49-46 lead at halftime. Mayo led the way with all 12 of his points. The Suns were hindered by shooting 43 percent and committing 11 turnovers.
Phoenix pulled even at 65 just past the midway point of the third on Hill's three-point play. Memphis countered with nine unanswered points, capped by a 3-pointer by Quinton Ross, to match its biggest lead.
That allowed Memphis to close the quarter with a 13-5 surge and carry a 78-70 lead into the fourth. Warrick had 10 in the quarter, while Barbosa scored 11 for the Suns.
The Grizzlies expected G Steve Francis, obtained in a Christmas Eve trade with Houston, to be ready for Monday night's game, but flight problems to Memphis delayed his arrival until later in the week. ... Nash missed only his fourth game of the season, and the first with back problems. Barbosa's start was his second of the season. The Grizzlies had lost their two previous games in overtime, 106-103 in double OT at San Antonio and 108-98 in Minnesota on Monday night. The Grizzlies have lost 10 straight in the series, with their last win over the Suns coming Dec. 17, 2005. Next on the career scoring list for O'Neal is Hakeem Olajuwon, who has 26,946 points. O'Neal's 24 points Tuesday night gave him 26,715.
Kevin Love, LeBron James and Kyrie Irving combine for 71 points as the Cavaliers go up 31 points at the half and rout the Raptors 116-78.
After the Raptors' Game 5 loss to the Cavaliers, Stephen A. Smith has some harsh criticism for Toronto, saying, "It was one of the most pathetic performances that I've ever seen in my life."
The Cleveland Cavaliers vaporized the Toronto Raptors in Game 5, smothering the Canadian backcourt to take a 3-2 series lead in the East finals.