Godfrey finished with 200 yards passing and 97 yards rushing.
The Tigers (2-7, 1-4 Conference USA) have now been shut out twice in C-USA play this season. They were dominated 42-0 by SMU late last month.
The Knights (4-4, 2-2) gained 505 total yards while holding the anemic Memphis offense to just 134. True freshman quarterback Taylor reed threw for 97 yards on 9-of-20 passing.
After seemingly finding their bearings on offense last week in a 33-17 win against Tulane, the Tigers had just six first downs against the Knights.
The Tigers never threatened after Godfrey scored on a 29-yard quarterback keeper with 12:41 left in the first quarter to cap a five-play, 70-yard drive.
The Tigers have off next week and will return to Liberty Bowl Memorial Stadium Nov. 12 against UAB.