Smith had a 4-yard scoring catch early in the first quarter, then followed it up with an 80-yarder in the second.
The Bulldogs (0-8, 0-6 region) tied the game early on a 63-yard pass from Daniel Woodard to Steven Mitchell. Woodard threw for another touchdown in the third period, this one to Steven Simerly for 10 yards.
But otherwise, it was all Yellow Jackets, and most of that came from Byrd. His 7-yard touchdown run before halftime gave Corner the lead for good. He added a 10-yard TD in the fourth quarter.
Austin Sanford, Hunter Doppler and Taylor Crawford also scored rushing touchdowns for the Jackets (5-3, 4-2), and Troy Elliott kicked a 44-yard field goal.
About the only thing that went wrong for Corner was their penalties — lots of them. The Jackets were flagged 19 times for 210 yards, which countered 475 yards of total offense.
Sanford ran 10 times for 154 yards, while Smith caught five passes for 122 yards and Dylan Hand caught five more for 85 yards.
Byrd was 13-for-21 for 219 yards in passing for Corner. Woodard was 13-for-20 for 193 yards, with an interception. Mitchell had eight receptions for 130 yards.
The Bulldogs fumbled the ball five times, turning it over on all five.
Corner travels to Bibb County to finish region play next week, while Carbon Hill heads to Dora.