I gave written notice today (december 3) of my 30 day notice to vacate to my landlord. I am on a month to month agreement. I was assuming I would pay this months rent and prorate January 1-3. He informed me that I am responsible for the whole month's rent for January; that he does not prorate rent. My lease agreement doesn't specify that notice must be given the 1st of the month, only that a 30 day's notice must be given. Is this just HIS policy or Missouri law? Who is in the right here?

Sorry, the landlord is correct.
http://www.moga.mo.gov/statutes/c400-499/4410000060.htm
4. (1) Except as provided in subdivision (2), the landlord or the tenant may terminate a month-to-month tenancy by a written notice given to the other party stating that the tenancy shall terminate upon a periodic rent-paying date not less than one month after the receipt of the notice.

This means that notice has to be given at least one month prior to the rent due date. Has you given notice on Friday, you would only owe for Dec, but since you gave notice after 12/1. the next rent date more than 30 days after giving notice is 2/1.