Veterans Back to Work Act of 2013 - Amends the Internal Revenue Code to: (1) make permanent the work opportunity tax credit for hiring qualified veterans (veterans receiving compensation for a service-connected disability and other federal assistance), and (2) allow employers who hire qualified veterans an exemption from employment and railroad retirement taxes for such veterans' first-year wages. Appropriates amounts to the Social Security Old-Age and Survivors Trust Fund and the Federal Disability Insurance Trust Fund to cover any revenue loss to such Funds resulting from this Act.