Deputies later learned Joseph Thomas, 23, had his 3-year-old son and the child's mother, Tiffany Willette, in the car during the chase.
Thomas eventually stopped the car in the 6100 block of Red Bridge Road and fled on foot with his son, deputies said. The child was in a full-body cast because he had a broken femur.
Sheriff's deputies are looking for Thomas. He faces charges of attempted vehicular assault, felony eluding, reckless endangerment, and 3rd degree driving with a suspended license.
The child was found safe and turned over to family.
Willette was arrested for rendering criminal assistance and resisting arrest.