Kaminsky, who also had 11 rebounds, scored on a layup, a hook shot and then a tip-in with one minute left in overtime for a 64-61 lead.
Point guard T.J. McConnell missed an 18-footer, but forward Aaron Gordon tipped the rebound back to McConnell. Guard Nick Johnson, the Pac-12 Player of the Year who ended up with the ball on the drive, was called for an offensive foul with 3.2 seconds left as Arizona coach Sean Miller looked on in disbelief.
There was a delay of several minutes as officials reviewed the ensuing inbounds play, as Jackson and Arizona forward Rondae Hollis-Jefferson battled for a ball that went out of bounds. Officials reversed the call to give the ball to Arizona.
Guard Gabe York gave the inbounds pass to Johnson, who dribbled to his left in search of a shot, but his 15-footer, which missed, came after the buzzer.
Kaminksy hit 3 of 5 3-point shots, showing off his inside-outside skills. Jackson was the only other Badger in double-figures with 10 points.
Arizona (33-5) was trying to reach its first Final Four since losing to Duke in the 2001 national title game. The Wildcats have lost their four regional finals and are 0-4 overall in regional finals in Anaheim.
Johnson led Arizona with 16 points. Hollis-Jefferson scored 10. Gordon had a career-high 18 rebounds to go with eight points.
With Arizona trailing 54-52 in the final minute of regulation and the shot clock at about 5, Johnson put up a contested 15-foot air-ball that Hollis-Jefferson grabbed and dunked with 30.9 seconds left. Wisconsin had the last shot, but Jackson missed a step-back 18-footer in the final seconds.
In a controlled game that was almost exclusively played in the half-court, neither team was able to gain big momentum swings, although Arizona put together a key 8-0 run after falling behind 41-36. Johnson capped the run with a 3-pointer for a 44-41 lead with 10:24 left.
Back came the Badgers, who went up 50-48 on Kaminsky's 3-pointer with 4:22 left, and the rest of the game played out as a tense grind.
Hollis-Jefferson tied the score at 50 on a contorted scoop layup, Wisconsin guard Josh Gasser scored on a tip-in and then Gordon responded with a reverse layup, his first field goal in nine tries, leading to the dramatic final minutes.
Neither defense gives up much easy -- and neither offense is turnover-prone -- so there was little in the way of the transition game in the first half. Arizona won the early battle of set offenses, never trailing and taking a 5-0 lead before Wisconsin forward Sam Dekker banked in a 3-pointer.
Arizona's lead was at 19-14 during a timeout with 7:59 left, when Wisconsin coach Bo Ryan was assessed a technical foul for unsportsmanlike conduct. Johnson made both foul shots and York hit one of two from the line for a 22-14 edge.