Ohio State won the Big 10 Conference title that year by going undefeated in conference play, while Minnesota lost one game (to Wisconsin) in 1961.
However, per an agreement the previous summer among the Big 10 Conference schools (athletic directors and presidents) , the plan was for the Big 10 to reject the Rose Bowl invitation at season's end. The purpose was to attempt to reduce the influence/importance of athletics at the college level.
After Ohio State declined the Rose Bowl invitation, the Rose Bowl committee then invited Big 10 runner-up Minnesota. The Gophers accepted, played and won (over UCLA) on 1/1/62...and the rest, as they say, is history.