Generally speaking, if the landlord can't prove that tenant's conduct caused the rats or vermin problems, it is the landlord's responsibility to take care of it. You might want to review Health and Safety Code 17923 which defines what a substandard building is and talks about the presence of certain pests as a habitability issue.
The landlord is absolutely responsible to ensure the rental unit is "substantially free" of rodents and vermin. See Civil Code sections 1941 and 1941.1. You are correct.