The Badgers let the game slip away in the final minutes with poor shooting and defense, giving Michigan State a surplus of opportunities to seal the game.
The Badgers, now 17-8, will host IL on Monday.
A Spartan perimeter defense that struggled in the first half held Badgers not named Happ to only eight second-half points. Wisconsin followed up the cold spell with a 14-4 run that featured seven points from Brevin Pritzl, five of which came in a single possession after a flagrant foul was called on Michigan State's Matt McQuaid.
Aaron Henry has the ability to make 3-pointers, and he has six makes on the season in limited shots, but hasn't made a single one since January 17 at Nebraska.
NEW YORK, NY - MARCH 24: Head coach Greg Gard of the Wisconsin Badgers looks on against the Florida Gators during the 2017 NCAA Men's Basketball Tournament East Regional at Madison Square Garden on March 24, 2017 in New York City. Following the review, Pritzl got two free throws, both of which he hit, and Wisconsin got the ball, which resulted in a Pritzl 3. Two jumpers from Wisconsin's Kobe King helped give the Badgers a 38-37 lead headed to the locker room.
Happ scored 20 points with 12 rebounds, and Nate Reuvers had 11 points for the Badgers (17-8, 9-5). Nick Ward finished with 12 points and six rebounds, while Aaron Henry added eight points - including a key jumper with 2:29 to play that sparked MSU's closing run.
"We have to be more complete when we get to the free throw line, get the ball to a high percentage area and be able to complete that". He was scoreless in the second half until a three-point play with 15.7 seconds left that sealed the victory.
Happ's 20 points put him at 2,000 for his career as he became the fourth player in the Big Ten - and the first in more than 35 years - to hit that mark and grab 1,000 rebounds.
In addition to the stout defensive showing, Michigan State out-rebounded Wisconsin 40-28.
Michigan State: The Spartans, who had lost three in a row to start February, have now won back-to-back games. "He made a couple of big, big plays tonight, that's for sure".
Wisconsin: The Badgers, who had their six-game winning streak snapped on Saturday by MI, missed out on a chance to keep pace with the conference leaders.
Next, Wisconsin will take on Illinois Feb. 18 at the Kohl Center. They will play four of their final six regular-season games at Breslin Center, with road games left at IN and MI.