On the first drive of the game, Mitchell Humphreys hit Ed Boehmer on a 32-yard TD pass to give Shawnee an early 6-0 lead.
Wyoming made it 7-6 lead on a 47-yard TD pass from Hauer to senior Dorien Fredericks midway through the second quarter.
Early in the fourth quarter, Meeks scored on a 2-yard run and converted the two-point conversion to make it 14-7. A minute later, Meeks scored on a 53-yard run to make it 20-7.
On its next drive, Hauer threw an 8-yard TD pass to Denham Miller to pull within six points.
With 2:14 remaining, Hauer found senior DJ Gray on an 8-yard TD pass and junior kicker Carter Rummer converted the extra point to make it 21-20.
The Braves went 1-for-9 on third down but converted 2-of-3 fourth down conversions. Shawnee lost three fumbles in the game.
The Cowboys (12-1) advanced to play fourth-seeded Kettering Alter (10-3) in the Region 16 final next Friday at a site to be determined.