DeAngelo Branche added 145 yards rushing and a touchdown and caught a touchdown pass for the Spartans (5-4, 4-3 Mid-Eastern Athletic Conference).
Norfolk State, which led the MEAC in total defense entering the game, gave up 300 yards, but held Morgan State (5-4, 3-3) to a third-quarter field goal after taking a 21-20 lead into halftime.
Brown's 16-yard touchdown pass to Branche put Norfolk State ahead 28-20 with 11:14 left in the third quarter.
Kemar Scarlett cut the lead to 28-23 with a 52-yard field goal with 8:14 left in the third quarter, but Justin Castellat added a 38-yard field goal for a 31-23 lead with 11:24 remaining in the game.
Norfolk State's P.J. Hayden had five catches for 109 yards.
Devan James led Morgan State with 77 yards rushing.