The council voted 11-to-1 Tuesday for the law, which would also require stores to charge 10 cents apiece for paper bags.
The ordinance, which requires a final vote next week, would take effect next year, starting with large stores and then expanding to small stores.
Supporters of the ban say plastic bags litter cities and beaches and endanger wildlife. Manufacturers say the ban would cost jobs.
Los Angeles County and such cities as San Francisco and Santa Monica already have bans in place.
Last month, the California Senate rejected a bill that would have phased out use of the bags in the state.