Soward scored on a pair of 1-yard runs, his last coming with 7:11 remaining to help secure Beechwood's 12th state championship.
Hazard quarterback Bailey Blair threw interceptions on two of the Bulldogs' final three drives.
Beechwood scored twice in the second quarter with Soward's 1-yard run and a 17-yard run from Aiden Justice to lead 14-7 at halftime.
Cory Smith put Hazard on the board first with a 7-yard run and finished with 128 yards on 29 carries.
Hazard (12-3) tied the game early in the fourth quarter with Blair's 9-yard TD pass to Braxton Whitaker, but Beechwood scored the game-winner on its ensuing drive.
Brayden Burch completed 16 of 26 passes for 162 yards for Beechwood (14-1).
