Mercedes Smith, 28, of Chicago died at Lutheran General Hospital, police said.
Glenview police responded to the 9800-block of Milwaukee Avenue just after 5:05 a.m. Tuesday for a report of a vehicle hitting a pedestrian.
The pedestrian, who was hit by a Cook County Sheriff's Office squad car, was taken to Lutheran General Hospital in critical condition, police said.
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Glenview police and the North Regional Major Crimes Task Force Major Crash Assistance Team are investigating.
It's not clear what led up to the crash.
The sheriff's office is cooperating with the ongoing investigation.
Anyone with information on the crash is asked to call the Glenview Police Tip Line at 847-901-6055.