Accountability of Taxpayer Funding for Afghanistan Fuels Act of 2013 - Prohibits a covered product from being provided directly or indirectly to the government of Afghanistan unless the Secretary of Defense (DOD) determines and reports to Congress that: (1) the covered product meets an existing or future need of the government of Afghanistan, (2) related financial documents are fully accounted for and will be maintained by DOD for at least 10 years, and (3) DOD has established related monitoring controls.

Prohibits DOD funds from being provided directly or indirectly to the government of Afghanistan to procure any covered product unless the Secretary determines and reports to Congress that: (1) the government of Afghanistan is capable of providing reliable accounting and reporting to DOD regarding the purchase, delivery, and consumption of a covered product; (2) the Special Inspector General for Afghanistan Reconstruction, or upon the date of termination of the Office of the Special Inspector General for Afghanistan Reconstruction, the Inspector General of DOD has certified to the Secretary that such requirements have been met.

Defines "covered product" as petroleum, oil, a lubricant (including diesel fuel, gasoline, jet or aviation fuel, kerosene, or other fuels), or firewood that is to be used for purposes of equipping, training, or sustaining the Afghan National Security Forces.