Dinwiddie was 14 of 15 from the free throw line as the Buffs (5-1) improved to 49-7 at home under coach Tad Boyle.
Xavier Johnson was back on the court for Colorado after sitting out two games with an undisclosed injury. He finished with nine points in 29 minutes.
The Gauchos were poised to pull off the upset, leading 54-52 with 5:51 remaining on a deep 3-pointer by Brown, one of seven he had on the night. But the Buffaloes went on a 12-0 run to take back control of the game.
The big play in the run was when Josh Scott was fouled on a layup and converted the three-point play to give the Buffs a lead they wouldn't relinquish. Those were the only points of the game for Scott, the team's leading scorer. Johnson also had a steal and a dunk to energize the crowd.
Williams went through the shoot around Thursday and was feeling better, but the team elected to give the 6-foot-7, 240-pound Williams more time to rest his balky back. Gone with Williams were 24 points and 13 rebound a game.
Santa Barbara also was missing John Green with a foot injury. The injury-plagued Gauchos did have Mitch Brewe back on the court after he sprained his left knee earlier this season in an exhibition game. He had four points and before fouling out.
Colorado led by seven midway through the first half, only to see the Gauchos roar back behind the torrid shooting of Brown, who hit 5 of 6 3-pointers to help his team take a 30-25 lead with 3:24 left before halftime. But the Buffaloes scored the last seven points of the half and went into the locker room with a 32-30 advantage.
Dinwiddie hit a bit 3-pointer in the first-half run to extend the Buffaloes' streak to 371 games with at least one 3-pointer.
This is the first of three games for Santa Barbara against Pac-12 schools. UCSB also travels to UCLA on Dec. 3 and hosts Cal three days later.
The Buffaloes are in the midst of playing six straight at Coors Events Center. They close out the home stand Sunday against Harvard.
Santa Barbara and Colorado are meeting for the first time since the Buffaloes' 76-55 win on Dec. 21, 1996, in Boulder. The Gauchos lead the all-time series 5-4.