Startup Innovation Credit Act of 2013 - Amends the Internal Revenue Code to allow a qualified small business, other than a tax-exempt organization, to use a portion of its tax credit for increasing research expenditures as an offset against its payroll tax liability under the Federal Insurance Contributions Act (FICA). Defines "qualified small business" as a corporation, partnership, or S corporation if the gross receipts of such entity for the taxable year are less than $5 million and such entity did not have gross receipts for any period preceding the 5-year period ending with such taxable year. Limits the amount of the payroll tax credit portion to $250,000 in any taxable year.