Calliste made the go-ahead jumper and free throw with 4:22 left and finished with 18 points for the Ducks, who trailed by as many as 12 points.
"I just didn't want to lose," he said. "Wasn't time to go home yet."
Johnathan Loyd added 16 points for the Ducks (22-8, 10-8 Pac-12) who greatly improved their chances of a bid in the NCAA tournament. Last season Oregon won the conference tournament and advanced all the way to the regional semifinals.
Aaron Gordon had 21 points for the Wildcats (28-3, 15-3), who had already clinched the Pac-12 regular-season title heading into next week's conference tournament in Las Vegas.
The Ducks won 13 straight to start the season and got ranked as high as No. 10, but then they lost five straight to tumble out of the poll and fall in the Pac-12 standings.
The Ducks considered the game a must-win after a six-game winning streak made them a possible bubble team for the NCAA postseason and gave them their fourth straight season with at least 20 victories.
As for the prospects following the victory over Arizona, Oregon coach Dana Altman wasn't going to speculate.
"I'm not going to worry about all that," he said. "We've got to go to the conference tournament and we've just got to play."
The Wildcats led by as many as 12 points in the first half, but Oregon closed to 31-29 at halftime. Mike Moser's 3-pointer pulled the Ducks to 36-34 early in the second half with a 3-pointer.
A six-point run put Arizona ahead 44-36 with 11:55 left and the Wildcats held on to the lead until Joseph Young's 3-pointer got the Ducks within 50-48 and Loyd's layup tied it at 50 with 5 minutes left.
Calliste hit a jumper and a free throw to give the Ducks a 53-51 lead with 4:22 left for their first lead since the game's opening minute.
Calliste and Loyd hit back-to-back 3s to stretch the lead to 59-51 and the 12,364 fans at Matthew Knight Arena were on their feet. The Ducks held on for the win and the crowd rushed the court.
Arizona had won five straight since a 69-66 overtime loss to rival Arizona State on Feb. 14.
"They were highly motivated," Arizona coach Sean Miller said about the Ducks. "With that, we had to play at our best and we didn't."
Oregon went into the game averaging a league-best 82.9 points a game, while the Wildcats were holding opponents to a league-low 58.5 points per game.
Arizona jumped out to a 9-2 lead that was snapped by Loyd's steal and fast-break layup for the Ducks. The Wildcats made it 20-8 on Nick Johnson's dunk midway through the first half.
Young's layup pulled Oregon within 24-18, but Caleb Tarczewski answered with a dunk for the Wildcats.
Oregon pulled to 31-29 at the half on Calliste's 3-pointer with 3-seconds to go before the break. Calliste made four of five 3-point attempts in the game.
Tarczewski led all players with 10 points in the first half.
"They're fighting for a tournament berth and guys are in zones," said Johnson, who finished with 11 points for the Wildcats. "Give him [Calliste] credit, he made the shots."
Moser finished with 10 points and 10 rebounds for his fourth straight double-double.
Arizona defeated Oregon 67-65 in Tucson back on Feb. 6. It was Arizona's first full game without sophomore forward Brandon Ashley, who is out for the season after surgery on his right foot.
Seven Oregon seniors were playing in their final regular-season home game.
Washington 82, USC 75 • C.J. Wilcox scored 24 points in his final regular-season game and Andrew Andrews added 19 as Washington held off a late charge from Southern California.
Wilcox, No. 2 on the school's all-time scoring list, was honored before the game as part of the Senior Day festivities. After a quiet first half, the Pleasant Grove High School graduate helped spark a big run to start the second that gave the Huskies a double-digit lead they hung on to at the end.
Oregon State 78, Arizona State 76 (OT) • In Corvallis, Ore., Roberto Nelson scored six of his 15 points in overtime, including the go-ahead free throw with 2:50 remaining, to pace Oregon State over Arizona State.
Jermaine Marshall's 3-pointer with 19.5 seconds to play in regulation time capped Arizona State's comeback from a 14-point deficit in the second half and forced overtime with the score tied at 67.
The Beavers (16-14, 8-10 Pac-12) earned a split of the two-game season series with ASU and snapped a four-game losing streak against the Sun Devils (21-10, 10-8) that had stretched over four seasons.
• Oregon 64, No. 3 Arizona 57
• Stanford 61, Utah 60
• Oregon State 78, Arizona State 76 (OT)
• Washington 85, USC 75
• Cal 66, Colorado 65 (OT)
• UCLA vs. Washington State, late