Marine Debris Emergency Act of 2013 - Amends the Marine Debris Act to: (1) require the Administrator of the National Oceanic and Atmospheric Administration (NOAA) to approve or disapprove a grant application with respect to a severe marine debris event and provide grant funds to an approved applicant within 60 days after receipt of the grant application, and (2) authorize the Administrator to give preference under the Marine Debris Program to projects that address such an event.

Authorizes state governors to request that the Administrator declare a severe marine debris event for purposes of grant consideration and disbursement. Requires the Administrator, within 30 days after receiving such request, to either declare such an event or provide a response explaining why the declaration was not made.

Allows the Administrator to: (1) require a grant recipient to provide training to persons engaged in grant-funded marine debris response efforts on the potential impacts of marine debris, including nonindigenous species related to the debris, on the economy, the marine environment, and navigation safety; and (2) enter into agreements to use the personnel, services, equipment, or facilities of federal agencies, state and local governments, regional agencies, interstate agencies, and other entities on a reimbursable or nonreimbursable basis.

Defines "nonindigenous species" to mean any species or other viable biological material that enters an ecosystem beyond its historic range, including any such organism transferred from one country into another.

Marine Debris Emergency Act of 2013 - Amends the Marine Debris Act to require the Administrator of the National Oceanic and Atmospheric Administration (NOAA): (1) to give preference under the Marine Debris Program to projects that address a severe marine debris event, including an event that has introduced or is likely to introduce marine invasive species to the United States; and (2) within 60 days after receiving a grant application with respect to such an event, to approve or disapprove such application and provide grant funds to an approved applicant.

Authorizes state governors to request that the Administrator declare a severe marine debris event for such purpose. Requires the Administrator, within 30 days after receiving such request, to either declare such an event or provide a response explaining why not.

Requires the Administrator to encourage grant recipients to: (1) educate staff and volunteers engaged in response efforts on the potential threats that marine debris and marine invasive species may pose, and (2) coordinate marine debris response efforts with appropriate entities.

Defines "marine invasive species" to mean a marine nonindigenous species the introduction of which to the United States will, or is likely to: (1) cause economic or environmental harm; (2) threaten the diversity or abundance of native species or the ecological stability of infested waters; or (3) disrupt commercial, agricultural, aquacultural, or recreational activities dependent on infested waters.