While the ordinance would prevent restaurants from automatically giving out straws, customers could still obtain them by request.
The council vote was not the final say on the matter, but it was an instruction to the City Attorney's Office to draft an ordinance that would be voted on early next year.
The council is also looking at the possibility of exploring a complete ban on plastic straws by 2021.
Council members argue the motion could help reduce the massive amount of plastic waste that goes into the city's landfills.
The Los Angeles measure goes a step further than a similar state bill signed into law by Gov. Jerry Brown earlier this year. The measure banned full-service restaurants from automatically giving out straws starting Jan. 1, 2019. The LA measure also applies to fast-food restaurants as well.
Under the state law, restaurants that don't comply will get two warnings before being fined up to $300 per year.