The incident began at 1:41 p.m. when officers saw two suspects, a male and female, approach two vehicles in the 1700 block of La Salle Avenue.
The male suspect allegedly stole one of the cars and, upon being stopped by the officers, rammed the police vehicle, then drove onto the sidewalk, struck a parked car and came to rest against a nearby home, police said.
The suspect tried to run away and allegedly dropped a firearm as he ran, but he was taken into custody shortly afterward, police said. His name was not immediately being released.
The female suspect drove away in a second stolen vehicle, which officers pursued briefly before stopping because of the risk to the public, according to police.
That vehicle was later recovered unoccupied in the city's Portola neighborhood and the female suspect remains at large.
No officers were hurt, but the home that was struck by the stolen car sustained some damage, police said.
Anyone with information about the case is asked to call the Police Department's anonymous tip line at (415) 575-4444 or to send a tip by text message to TIP411 with "SFPD" in the message.