The Hornets drove downfield, but Preston Clark missed a 30-yard field goal with :12 remaining.
The Bulldogs trailed until they put together a 95-yard drive capped by the scoring toss to McKenzie for the win.
Trailing 27-14, A&M (3-5, 2-3) closed the gap when Lee, who threw for 319 yards and three touchdowns, hit Montaurius Smith with a 20-yard scoring strike. Nine plays later Alabama State's Malcolm Cyrus ran in from the 5 for a 33-21 lead.
Brandon Eldemire scored early in the fourth to close to 33-28. Clark followed with a 43-yard field goal to give Alabama State (4-4, 3-3) a 36-28 lead with 11:32 left. Ceasar Diaz-Ramon then kicked a 40-yarder to cut the lead to 36-31.