After trailing for two quarters, Wyoming (1-0, 0-0 Mountain West) built a 13-play, 84-yard drive, capped by Crum's 28-yard touchdown pass to Donate Morgan to give the Cowboys the final margin at 7:28 in the fourth.
Wyoming's defense held Ohio scoreless for the rest of the game.
Ohio's Theo Scott went 26-of-34 for 224 yards, one touchdown and one interception.
Ohio (0-1, 0-0 Mid-American) had gone up 20-14 with 13:45 to play on a 34-yard field goal by Barrett Way.
Ohio led 17-14 after a back-and-forth first half.
The Bobcats scored on their first possession, driving the ball 59 yards over seven minutes. Scott found Andrew Mooney with a 4-yard TD pass to put the Bobcats up 7-0.
Wyoming answered with a 51-yard drive, capped by a 6-yard TD run by senior running back Devin Moore to tie the score at 7.
Scott fumbled a snap on Ohio's first possession of the second quarter. Wyoming recovered to set up a 61-yard drive and a score as Crum found junior Jesson Salyards on a 3-yard TD pass.
Ohio's Donte Hardin grabbed the ensuing kickoff and returned it 100 yards to tie the score at 14.
Ohio's Steven Jackson intercepted Crum on the next drive. Way's 31-yard field goal put the Bobcats up 17-14 with about 4:30 to play in the half.
Crum was intercepted again on the next possession -- this time by Ohio senior strong safety Neil Jereb.