The crash happened a minute into the pursuit at 85th and San Pedro streets around 9:10 p.m.
The driver of the gray BMW being pursued hit the curb while turning westbound on 85th Street, and the officer following close behind lost control, crashing into an emergency shelter for women and children.
No one inside the shelter was injured, but the building was red tagged due to extensive damage. Authorities said 11 adults and 21 children have been displaced.
The two officers inside the vehicle were taken to a hospital. One has since been released. The second officer remains in the hospital with a fractured ankle.
The suspect is still at large, but investigators have found the BMW.
Anyone with information was asked to call the Los Angeles Police Department's Newton Station at (323) 846-6547.