This is another legal question that would require an attorney to review documents and answer. In the alternative, there might be some local landlord/tenant entity that would help with advice. I would only say that whatever you pay, every month, make sure you obtain and keep receipts.
No deposit was required. That's OK. A deposit is at the option of the owner. Depending on how the lease is written, it might or might not cover missed rent payments. But you weren't required to pay.
You say the "last month's rent" was not included in the contract. Please clarify. Usually a lease will state a term (such as 1 year from January 1) and either a total amount (such as $12,000, which works out to $1,000 a month) or a monthly amount (such as $1,000.
It's very likely that your lease was worded one way or the other. If you were in the property for the duration of lease--for the time span of the lease--then you owe the money. The only exception that I've run into--and this happens with apartment rentals mostly--is when a free month is explicitly stated. That's usually an incentive offered by the landlord: "Sign a 12 month lease and your 13th month is free" backed up by that language in the lease.
But, absent that, you not only rented during that month but--unless you can quote the exact language from the lease absolving you from payment--owe the rent.