Using forcible stopping tactics, the officers were able to stop the vehicle and apprehend two suspects.
JSO said a third suspect, described as a thin black man with short dreadlocks, fled the scene.
Police said he was wearing a brown sweatshirt.
An officer was injured during the pursuit and was taken to the hospital with non-life threatening injuries. The man shot was also taken to the hospital with non-life threatening injuries.
"The circumstances around this incident are unknown at this time and the case will be investigated by the Aggravated Battery Unit. We encourage anyone who may have information on this incident to contact the Jacksonville Sheriff's Office or Crime Stoppers at 866-845-TIPS," JSO said.