Bayless, a freshman from Phoenix, has led the Wildcats in scoring in three of his first six college games.
Shooting 62.1 percent from the floor, Arizona thrashed the Titans in a break between games against Top 10 teams. The Wildcats lost in overtime at No. 4 Kansas last weekend and play host to No. 9 Texas A&M on Sunday.
The Wildcats (4-2) wasted little time putting away Cal State Fullerton (3-1), bolting to a 20-4 lead as Budinger scored eight of his team's first 10 points.
It was Arizona's sixth game under interim coach Kevin O'Neill, who replaced Lute Olson when the Hall of Famer took a personal leave of absence. On Monday night, Olson appeared at his first practice since going on leave three weeks ago but he's not expected to coach in a game soon.
Still searching for the right combination, O'Neill started his fourth different lineup in six games, with forward Brett Brielmaier replacing Kirk Walters.
O'Neill has stressed defense since taking over for Olson, and the Wildcats played unusually hard at the defensive end from the opening tip.
Arizona blocked four shots in the first four minutes and limited the Titans to 37.5 percent shooting from the floor in the first half. Cal State Fullerton had averaged 85.3 points in its first three games.
Meanwhile, Arizona's offense was firing from the start. The Wildcats shot 66.7 percent from the floor in the first half, hitting 6 of their first 7 (85.7 percent) from beyond the arc.
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Former Kentucky guard Ed Davender, the only Wildcat with at least 1,500 points and 400 assists, has died. He was 49.
Guard Naz Mitrou-Long, whose 2015-16 season ended in December due to hip problems, has been granted a hardship waiver for the coming season, Iowa State said.
As the Sooners look to life without superstar Buddy Hield and his sidekicks, it's clear their unseasoned players will need to grow up fast.