CERTIFICATION PURSUANT TO RULE 30A-2(B) UNDER THE 1940 ACT AND
                     SECTION 906 OF THE SARBANES-OXLEY ACT

I, James A. Bowen, Chairman of the Board,  President and Chief Executive Officer
of First Trust Enhanced Equity Income Fund (the "Registrant"), certify that:

         1.     The Form N-CSR of the Registrant  (the "Report")  fully complies
                with  the   requirements  of  Section  13(a)  or  15(d)  of  the
                Securities Exchange Act of 1934, as amended; and

         2.     The information  contained in the Report fairly presents, in all
                material  respects,  the  financial  condition  and  results  of
                operations of the Registrant.

Date:    2/26/09                                /S/ JAMES A. BOWEN
     ----------------------             ----------------------------------------
                                        James A. Bowen, Chairman of the Board,
                                        President and Chief Executive Officer
                                        (principal executive officer)


I, Mark R. Bradley,  Treasurer,  Controller,  Chief Financial  Officer and Chief
Accounting   Officer  of  First   Trust   Enhanced   Equity   Income  Fund  (the
"Registrant"), certify that:

         1.     The Form N-CSR of the Registrant  (the "Report")  fully complies
                with  the   requirements  of  Section  13(a)  or  15(d)  of  the
                Securities Exchange Act of 1934, as amended; and

         2.     The information  contained in the Report fairly presents, in all
                material  respects,  the  financial  condition  and  results  of
                operations of the Registrant.

Date:    2/26/09                                /S/ MARK R. BRADLEY
     ----------------------             ----------------------------------------
                                        Mark R. Bradley, Treasurer, Controller,
                                        Chief Financial Officer and Chief
                                        Accounting Officer
                                        (principal financial officer)