Military Pay Continuation Act of 2013 - Requires the Secretary of the Treasury, during a government-wide shutdown, to transfer to the Secretary of Defense (DOD), and the Secretary of Homeland Security (DHS) in the case of the Coast Guard, amounts necessary to continue to provide pay and allowances to members of the Armed Forces, including the reserves, who perform active service during the funding gap. Allows for the provision of such pay and allowances, at the discretion of the Secretary of Defense, for: (1) civilian DOD personnel providing support to such members, and (2) DOD contractor personnel providing direct support to such members.