Coming off its first loss of the season, Arizona (22-1, 9-1 Pac-12) labored offensively in its first game without Ashley, who is out for the season with a foot injury.
As they have so many times this season, the Wildcats pulled out a victory by turning up the defensive pressure late to overcome a 4-for-16 effort from 3-point range and 16 missed free throws.
T.J. McConnell hit a 3-pointer with 90 seconds left and Johnson went 5 of 6 from the free throw line in the final minute to prevent the Wildcats' first losing streak of the season.
Oregon (15-7, 3-7) kept up with Arizona and led late in the second half, but stumbled down the stretch.
Joseph Young and Jason Calliste had 14 points each and Damyean Dotson added 12 for the Ducks, who missed key free throws in the closing seconds and a desperation 3-point attempt at the buzzer.
Forced to shuffle his lineup with Ashley out, Arizona coach Sean Miller moved Hollis-Jefferson from his sixth-man spot into the starting lineup and long-range shooter Gabe York slid into his first-off-the-bench role.
The new-look rotation was solid early, dropping in 3-pointers and flying in for dunks while Kaleb Tarczewski had his way inside.
The Ducks started double-teaming Tarczewski later in the half and McConnell picked up his third foul late, upsetting the Wildcats' offensive rhythm. It didn't help that Arizona struggled—again—from the free throw line, making 11 of 19 in the first half.
Arizona continued to labor offensively to open the second half, missing seven of its first eight shots as Oregon pushed the lead to 46-40.
The Wildcats pulled within 60-59 on a putback by Tarczewski. McConnell, who missed seven of his first eight shots, put Arizona ahead on 3-pointer with 1:30 left and Johnson made it 64-60 with two free throws.
Oregon, the defending Pac-12 tournament champion, has been in a downward spiral since conference play started.
Ranked as high as 10th, the Ducks will likely need to start stringing together wins to get back to the NCAA tournament.
Oregon opened the Pac-12 with an overtime win over Utah, but followed with five straight losses before winning two of its past three, the latest a 78-66 win over USC behind a Matthew Knight Arena-record 16 steals.
The nation's fourth-best scoring team, the Ducks had a hard time getting good looks against the long, athletic Wildcats, missing 11 of their first 15 shots.
Oregon started to find the range late in the first half behind Calliste (11 points) and Dotson (10), taking a 38-37 lead.
After some early struggles, the Ducks started dropping in shots midway through the second half, building a 50-43 lead.
Even when Arizona rallied by turning up the defensive pressure, Oregon had an answer, keeping just ahead of the Wildcats until the closing minutes.
But the Ducks couldn't hold the lead, falling behind by four after a shot-clock violation and a wild 3-point attempt by Young.
Oregon pulled within 64-62 on Dominic Artis' shot in the lane, but had a string of missed free throws and turnovers before missing on a final 3-point attempt.