The Skyhawks (4-5, 3-3 Ohio Valley Conference) took a 21-0 lead on Young's second touchdown, a 1-yard run with 5:12 left in the fourth quarter.
Young, who had 32 carries, also scored on a 4-yard run to give Tennessee-Martin a 7-0 lead in the first quarter.
The Skyhawks held Tennessee State (3-6, 2-3) to 269 yards of offense and the Tigers were 2-for-16 on third-down conversions.
Tennessee-Martin's Cade Thompson was 11-for-22 for 115 yards and scored on a 1-yard run for a 14-0 lead with 7:19 remaining in the second quarter.
Tennessee State's Calvin McNairl was 8-for-13 for 134 yards including a 73-yard touchdown pass to Julian Sample. Dominic Grooms was 9-for-20 for 73 yards. McNairl and Grooms combined to throw three interceptions.