It depends upon your state's law. The general rule would that the contract (aka the lease agreement) terms govern. So if it has a clause saying it will auto-renew if you dont give at least 60 days notice, then it renews if you dont give the notice. HOWEVER, your state may give tenants more protection for public policy reasons. You will need to contact local counsel to review the specifics of your lease.
Have you gone to the landlord and attempted to negotiate with him? I am assuming you missed the notice deadline and want to leave. My landlord clients typically will release the tenant from the autorenewal of the lease in exchange for X (usually 1 month's rent) and the unit being left in excellent condition.
Washington landlord tenant relations are generally governed by the Residential Landlord Tenant Act. A landlord may not require a tenant to waive or forgo a right that a tenant has under the RLTA unless specific conditions are met (RCW 59.18.230).
Under Washington law, a tenant can give a notice to vacate a late as 20 days on a month to month tenancy prior to the end of a rental period. Any requirement in a lease to lengthen that time period would be infringing upon the right of the tenant to give notice within the time frame outlined in the RLTA.
Your lease will state what is to happen after the end of the initial term. The lease may specifically end if no renewal is given, or it may continue on with a month to month basis. If it continues with a month to month basis you should only be required to give a 20 day notice.