Alabama shot just 21 of 59 (35.6 percent) from the field, including 4 of 17 on 3-pointers.
The Crimson Tide led 26-24 at halftime following a buzzer-beating jumper by Cooper to end the first half.
Retin Obasohan extended Alabama's lead to 30-24 early in the second half with a layup, but Portis followed by continuing his hot play of the first half.
The freshman scored Arkansas' first 11 points of the second half, making it 17 straight points overall, to put the Razorbacks up 35-33.
"Bobby Portis, tonight, was the story of the game," Alabama coach Anthony Grant said. "For a freshman, to able to do what he did tonight ... He was outstanding."
Portis had 29 of Arkansas' first 35 points, and he was 12-of-15 shooting at that point — compared to just 1 of 25 for the rest of his teammates. He passed the previous mark of Thurman — now a member of the Arkansas support staff — with a pair of free throws with 18.6 seconds remaining, giving the Razorbacks a 63-58 lead and putting the game out of reach.
"(Thurman) will be pretty mad about it, I know he will," Portis joked.
Alabama's last lead came on a pair of free throw by Cooper to go up 46-45.
Alandise Harris, who was suspended for Arkansas' last game at LSU, followed with a jumper to put the Razorbacks up for good. The points capped a run of eight straight points for the junior, who scored all of his points during the stretch.
Both teams entered having lost four of their last five games, with the Crimson Tide's losses coming by an average of 15 points.
Alabama still hasn't won away from home in nearly a year. Its last road win was a 52-45 win at Georgia, on Feb. 12, 2013, and it's 0-12 away from home since then.
"Their defense limited us offensively in terms of what we were able to do," Grant said. "We weren't able to have much success offensively."