As a senior, he helped his team earn a 10–1 record and the East Shore Athletic League championship.

As a redshirt freshman in 1991, he was the third-string quarterback behind Billy Joe Hobert and Mark Brunell, on the national championship team.

As a junior, he returned to quarterback, as the backup to Damon Huard, getting a chance to start the final 3 games of the 1993 season, registering a 2–1 record.

As a senior, he was moved back to wide receiver, leading the team with 49 receptions (at the time fifth highest total in school history), for 770 yards and 7 touchdowns.

Bjornson was selected by the Dallas Cowboys in the fourth round (110th overall) of the 1995 NFL draft,[2] with the plan of converting him into a tight end.

He became a starter in 1996, after Pro Bowler Jay Novacek suffered a back injury, that eventually forced him to retire.