Kyle Davis added 15 points for USU (14-9, 4-7 Mountain West Conference) and Spencer Butterfield 13.
Tied at the break, the Aggies shot 63 percent in the second period and never trailed after Medlin landed a 3-pointer just seconds into the second half. Ryan Watkins tied the game at 43, but inside buckets from Shaw and a 3-point shot by Medlin helped extend the USU lead to 56-47 with 10:10 remaining.
Boise State led most of the first half until USU's Davis tied the score at 33 with two free throws leading into intermission.
Anthony Drmic scored 26 for Boise State (15-9, 5-6) and Derrick Marks added 21 points.
Utah State has won 18 straight at home against the Broncos, but split the series this season.