Scott had a 20-point lead in the second quarter and was up 43-31 at the half.
Campbell County, which shot 76 percent from the field in the second half, rallied and took its first lead at 65-64 with 4:21 left in the game.
The Camels were up 72-67 with 2:30 to go, but Scott made back-to-back layups off steals to get to within a point.
Campbell County point guard Joel Day hit two free throws to push the lead back out to three.
After Vincent Dumlao scored for Scott, Day made two more free throws and the Camels were up 76-73 with 30 seconds left.
Nelson Perrin made a layup for Scott with 20 seconds to play.
Cole Hegyi made one of two free throws to give Campbell County a 77-75 lead with 13 seconds left.
After a timeout, Scott drove into front court and Jaycob Pouncey missed a long three-pointer.
Ohmer ran down the rebound, drove into the right corner and shot the game winner.
The ball bounced on the rim and fell through, giving Scott its first 10th Region championship.
The Eagles won the 9th Region championship in 1989.
Ohmer scored 21 of his game-high 30 points in the first half.
Hegyi led Campbell County in scoring with 20 points.
Scott will play 13th Region champion Harlan County in the first round of the Whitaker Bank/KHSAA Boys Sweet 16 Wednesday, March 14 at 6:30 p.m. at Rupp Arena.
S (21-12) C. Ohmer 12, J. Ohmer 30, Jolley 13, Perrin 12, Pouncey 1, Dumlao 10
CC (30-5) Day 17, Bertsch 12, Wilson 3, Hegyi 20, Clos 13, Jolly 10, Geiman 2
S 23 20 15 20- 78
CC 13 18 20 26- 77