After watching Youngstown State rally in the first half to earn a 14-all halftime tie, Missouri State (2-3, 1-1 Missouri Valley Conference) scored on its first two possessions of the second half.
Johnston scored first for Missouri State in the second half on a 38-yard run with 11:58 left in the third quarter followed by a 5-yard touchdown run by Kirby.
Youngstown State's Jabari Scott scored his second touchdown of the game with 29 second left in the third quarter on a 1-yard run to narrow Missouri State's lead to 28-21.
Missouri State responded with a touchdown of its own on a 12-yard pass from Kirby to Jeff Sergant to go up 35-21.
Washington rushed for 133 yards and Kirby threw for 127 yard on 11 of 14 passing as Missouri State finished with 399 yards of offense.
Kamryn Keys had 113 yards rushing for the Penguins.