Osbourne's touchdown with 3:23 left put the Great Danes (7-5, 3-5 Colonial Athletic Association) up by 10 less than two minutes after the Seawolves (5-7, 4-4) pulled within 20-17 on Marcus Coker's 2-yard TD run.
Osbourne finished with 167 yards on 24 carries. Will Fiacchi threw for 135 yards, a touchdown and an interception.
Albany led 13-3 going into the fourth quarter. Kinder threw a 79-yard touchdown pass to Adrian Coxson to make it 13-10.
Albany answered with an 11-play, 76-yard drive, capped by Josh Gontarek's 13-yard touchdown run.
Kinder completed 7 of 10 passed for 178 yards, a touchdown and two interceptions. Coxson had six catches for 117 yards.