Johnson completed 14 of 20 passes for 219 yards and added 29 yards rushing.
Guy Lemonier's 8-yard touchdown run capped a 12-play, 93-yard drive and gave Howard (1-7, 1-5 Mid-Eastern Athletic Conference) a 7-0 lead midway through the first quarter.
Johnson hit Myles Williams for a 47-yard score and Ricquaz Brannon for a 21-yard touchdown to extend the Bison's lead to 20-0 going into the second quarter.
Savannah State (1-6, 1-4) committed four turnovers, two of which were returned for scores.
Travon Hunt had a 31-yard fumble return for a touchdown that gave Howard a 41-9 lead late in the third and Kenneth Russ had an 11-yard pick-6 - just 13 seconds after Johnson's 1-yard scoring run - to cap the scoring with 9:31 to play.