After winning their first 15 games of the season, the Buckeyes have lost consecutive contests to Michigan State, Iowa and Minnesota.
The 6-11 Eliason controlled the paint, connecting on 6-of-9 field goal attempts for 12 points and claiming 14 rebounds.
Ohio State had a one-point lead with 13 minutes to go, but the Buckeyes' LaQuinton Ross missed a technical foul free throw attempt and then failed to connect on a 3-point try.
Minnesota proceeded to go on a 7-0 run to take the lead for good. Deandre Matthieu led Minnesota in scoring with 13 points and Andre Hollins added 10.
The Gophers (14-4, 3-2 Big Ten Conference) made just 3-of-14 from 3-point range, but hit 50 percent from the field overall.
Ohio State (15-3, 2-3 Big Ten) was led by 22 points from Ross.