First Trust Energy Infrastructure Fund (FIF) N-CSR Certified annual shareholder report (management investment company)

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 Energy Infrastructure 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:	February 8, 2024	/s/ James M. Dykas
		James M. Dykas, President and Chief Executive Officer (principal executive officer)

I, Derek D. Maltbie, Treasurer, Chief Financial Officer and Chief Accounting Officer of irst Trust Energy Infrastructure 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:	February 8, 2024	/s/ Derek D. Maltbie
		Derek D. Maltbie, Treasurer, Chief Financial Officer and Chief Accounting Officer (principal financial officer)