Officials say all three suffered non-life-threatening injuries and are in stable condition.
The crash happened Sunday just after 4 p.m. in the 1000 block of S. Main Street, near the entertainment complex L.A. Live, officials said.
The L.A. Fire Department, who responded to the scene, initially reported that the crash happened at the end of a pursuit. LAPD later said the officers were actually responding to a report of a man with a gun.
2 LAPD officers injured in car crash after chase ends in downtown LA
As the officers crossed the double yellow line to try and get around traffic with their lights and sirens on, the agency said a van made a left turn and collided with the police vehicle.
Two other people declined medical treatment following the crash, according to LAFD.
No further details were immediately known.