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

I, James M. Dykas, President and Chief Executive Officer of First Trust
Exchange-Traded AlphaDEX(R) 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: October 9, 2020                   /s/ James M. Dykas
     -----------------                  ----------------------------------------
                                        James M. Dykas, President and
                                        Chief Executive Officer
                                        (principal executive officer)


I, Donald P. Swade, Treasurer, Controller, Chief Financial Officer and Chief
Accounting Officer of First Trust Exchange-Traded AlphaDEX(R) 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: October 9, 2020                   /s/ Donald P. Swade
     -----------------                  ----------------------------------------
                                        Donald P. Swade, President and
                                        Chief Executive Officer
                                        (principal executive officer)