It was Northwestern's first win over the Badgers in Madison since 1996.
The smooth Crawford glided around the court for tough shots - none harder than a 3 off a curl with Wisconsin's Josh Gasser's hand in his face to give Northwestern (11-11, 4-5 Big Ten) a 13-point lead.
Wisconsin (17-4, 4-4) turned up the pressure late to get within six with 41 seconds left. But Traevon Jackson lost the ball on a drive with 23 seconds left to seal the Badgers' fate.
Ben Brust's 21 points led Wisconsin, which lost for the fourth time in five games.
After a 16-0 start to the season, defense has been a problem for the Badgers. Coach Bo Ryan's club had their issues again, especially during an 8-0 run in the second half by Northwestern that broke a 34-34 tie.
But offense was the bigger problem for Wisconsin, which uncharacteristically settled for some bad looks against Northwestern's solid defense.
Crawford hit an off-balance layup with the shot clock winding down for a 54-39 lead with about 3:30 left.
The Badgers used pressure to get within 62-56 after Nigel Hayes went 1 of 2 from the foul line.
The Wildcats, though, held on from there and Jackson's turnover proved costly.
Crawford finished 10 of 15 from the field and added eight rebounds, while Tre Demps scored 10 points.
Sam Dekker finished with 11 points and seven rebounds for Wisconsin, but was 2 of 9 from the field.
The Badgers overall finished 15-of-57 shooting, and hit more foul shots, going 21 of 27 from the line. They were 5 of 24 from 3-point range.
Northwestern first-year coach Chris Collins earned his first big win on the road, and the Wildcats got their first victory over Wisconsin overall since 2009. It was a sharp contrast to teams' first meeting this year, a 76-49 Badgers blowout in Evanston earlier this month.
But things have changed for Northwestern since then. They're 4-2 in their last six games, and had held opponents to an average of 51 per game prior to a 76-50 loss last weekend to Iowa.
Crawford, a fifth-year senior who has played a school-record 132 games, showed his savvy all night. Often playing against one of the Big Ten's best defenders in Gasser, Crawford repeatedly made clutch shots down the stretch.
With 8:19 left, Gasser played nearly-perfect defense, but the 6-foot-5 Crawford jumped high to hit a long bucket, pumping his right arm and letting out a yell as he turned back up the court to play defense.
Wisconsin shot just 26 percent in the first half, but got back into rhythm briefly to start the second after getting the ball into the lane.
Frank Kaminsky added eight points and 10 rebounds, including two offensive boards early in the second half that helped the Badgers briefly get out of an offensive funk.
Kaminsky slapped an offensive rebound to a teammate in the backcourt, then got another board on the same possession that led to a long 3 by Brust for a six-point Wisconsin lead.
Crawford had the answers down the stretch for the Wildcats.
Crawford scored 10 in a first half that had Northwestern trailing 23-22. The Badgers shot 6 of 24 (25 percent) in the first half, including an 0-for-10 stretch.