Northwestern won on the Badgers' home floor for the first time in 18 years.
Wisconsin held a 32-27 lead with 15 minutes to play and Northwestern outscored the Badgers 22-5 over the next 9:30.
Crawford finished with 30 points and eight rebounds, lifting Northwestern to 10-10 for the season and to 4-5 in the Big Ten Conference.
Tre Demps and JerShon Cobb added 10 points each for the Wildcats.
Wisconsin (17-4, 4-4 Big Ten) was led by Ben Brust with 21 points. The Badgers made only 15-of-57 shots from the field and went 5-of-24 from long range.