The Eagles jumped out to a 17-7 lead after the first quarter, but the Rats closed the gap, tying the game 31-31 after the third.
FCC outscored Holland Black River 12-10 during the fourth quarter, including the winning three-point shot by Byers. The Rats missed a 3-point attempt with two seconds left.
Will Dukes led the Eagles with 18 points while Nick Roberts had 10 points and six rebounds. Alec Cammenga pulled down seven rebounds.
The Eagles are now 2-1 on the season and will take on Kent City Algoma Christian on Monday.