Annexstad, a Minnesota native who attended high school at IMG Academy in Florida, beat out redshirt freshman Tanner Morgan, the only scholarship quarterback on the roster, to win the job.
Annexstad fielded some FBS recruiting interest -- including a scholarship offer from Cincinnati -- but decided to return home to Minnesota, where he enrolled prior to spring practice as a preferred walk-on.
Fleck told reporters he does not expect to rotate between the two quarterbacks.
"This does not mean that you won't see Tanner Morgan in a game," Fleck said, according to the Pioneer Press. "I can't say that. He could possibly be in, but this is again not a rotation. This is not going to be a short leash for somebody. At some point, we've got to be able to sit there and say, 'Let's give this guy a lot of opportunities.'"
Annexstad's father, Scott, also played for Minnesota.
Conor Rhoda and Demry Croft shared the quarterback job last season for Minnesota. But after Rhoda graduated and Croft transferred to Tennessee State, the Gophers were left without any quarterbacks with FBS game experience.
That glaring lack of depth was addressed by the arrival of junior-college transfer Victor Viramontes, but he only spent a short time at Minnesota before returning to junior college in California.
Only eight teams in the FBS finished with fewer passing yards per game than Minnesota in 2017.
Information from The Associated Press was used in this report.