Military Savings Act of 2013 - Requires the Under Secretary of Defense (DOD) to carry out a five-year pilot program at at least 10 military installations to develop innovative consumer financial products that encourage savings and wealth-creation among members of the Armed Forces on active duty.

Authorizes products designed to: (1) increase the rate of savings among such members by providing automatic deposit of special pay and allowances; (2) reduce the need for high-cost short-term lending services by providing alternatives, such as financial institutions providing an option for members to receive advances on salary payments; and (3) address obstacles to traditional consumer banking and lending for members with limited credit history.

Requires a financial institution seeking to begin or renew operating on a military installation selected for such program to agree to offer such products and to notify members about the availability of such products.

Directs the Under Secretary to contract for an annual independent evaluation of such program.

Authorizes the Under Secretary to extend the program and expand it nationwide upon determining that such expansion would improve member savings rates or decrease their need to rely on payday lenders.