Koback's 1-yard score tied the score at 21 early in the fourth quarter, capping a nine-play, 70-yard drive.
With Wilson out of the game in the final minute, backup Jaren Hall directed the Cougars to the Toledo 32 before his pass went beyond the end zone on the game's final play.
BYU coach Kalani Sitake said after the game that Wilson suffered an injury to his throwing hand and expected the quarterback to miss some playing time.
Mitchell Guadagni threw for 206 yards with a touchdown and an interception for Toledo (3-1), which has won three straight.
Wilson was 22 of 38 for 315 yards and two scores, both to Aleva Hifo. Hifo caught five passes for 111 yards and two scores including a 75-yarder for BYU (2-3).
Robert Mathis, Erik Williams selected for Black College Hall of Fame
Robert Mathis of Alabama A&M, Erik Williams of Central State University and coach Joe Taylor have been selected for induction into the Black College Football Hall of Fame.
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College Football Playoff: The best, worst and most realistic conference scenarios
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'Stanford Steve' and 'The Bear': Week 12 college football picks, bets, nuggets
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Best bets for Week 12 college football games
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When a Georgia football game was interrupted by my dog on the field
Mark Schlabach tells the infamous story of when his dog ran onto the field during the 1997 Georgia-Kentucky game.