Kaminsky had 23 points and 11 rebounds, and the fifth-ranked Badgers won 68-49 on Sunday to clinch the best start in school history.
Kaminsky hit eight of his first nine shots on his way to 17 first-half points, but was then held scoreless for more than 16 minutes. But Koenig and Hayes combined to score 23 points in the second half as Wisconsin (23-2, 11-1 Big Ten) put the game away.
"Our guys really don't go out there and say, 'OK, let's make sure Frank gets his X number of points and then we'll go get ours,'" coach Bo Ryan said. "We took what they gave us. That's all we ever do."
Koenig finished with 15 points, and Hayes added 14.
Illinois (17-9, 7-6) started the game hot, and four 3-pointers in just more than three minutes infuriated Ryan. Wisconsin responded by clamping down on defense.
The Illini then missed seven of their last eight shots in the half, and the Badgers sandwiched a 20-4 run around halftime. Illinois got no closer than nine points the rest of the way.
"It just goes to show you that we can beat you in a lot of different ways," Kaminsky said.
The Illini had held all four opponents to under 60 points and 40 percent shooting. But of those teams, only Michigan State ranks in the top half of the Big Ten in scoring average.
Illinois coach John Groce said he was most disappointed that his team was outrebounded 32-25 and allowed Wisconsin to turn 15 offensive rebounds into 12 points.
"Now you've got to sit down and guard them again," Croce said. "They're already the No. 1 offensive efficiency team in the last 17 years of analytics, so it would be preferable to guard them one time, instead of multiple times in a trip down the floor."
Illinois: The last time the Illini defeated a top-five team on the road was March 5, 1989, when Nick Anderson's 35-footer at the buzzer led Illinois past No. 3 Indiana 70-67. ... Rice had averaged 21.5 points and nine rebounds in his games against Wisconsin last season.
Wisconsin: Josh Gasser will need at least one more game to become only the second player in school history with 1,000 points, 500 rebounds and 250 assists. He came into Sunday needing 10 points to accomplish the feat but was held to three points on 1-of-5 shooting. Michael Finley finished his Wisconsin career with 2,147 points, 648 rebounds and 371 assists.
RICE STILL FINDING STROKE
Rice was a bit better Sunday than he was Thursday against Michigan, his first action after missing nine games because of injury and suspension. Rice, who wore a wrap on his non-shooting hand, was 4 of 12 from the field, including 1 of 5 from the 3-point line. In Thursday's win, he had four points on 2-of-7 shooting.
RYAN ON HALL HONOR
Ryan received a standing ovation before the game for being selected as a finalist for the Basketball Hall of Fame. Asked what the honor would mean, he said it would be a "thank you" to all those who he's played for, coached and worked with, including administrators and faculty. Ryan said he's received a number of well wishes from former players. "If that would put the smile on the face of the 12th man that I had at Brookhaven Junior High School, Sun Valley, Platteville, Milwaukee, Madison," Ryan said before pausing, "I'd be pretty happy. I'd be real happy."
Illinois hosts Michigan State on Feb. 22.
Wisconsin visits Penn State on Wednesday.
Mike Brey is looking at former Notre Dame players to fill two assistant jobs that have opened up with the departures of Martin Ingelsby and Anthony Solomon.
Purdue's 2016-17 prospects got a jolt Wednesday when forward Caleb Swanigan announced he will return for his sophomore season.
Clemson's Jaron Blossomgame says he is withdrawing from the NBA draft and returning to school for a final season.