Esposito connected with Jason Poore on a 34-yard touchdown pass to give Albany (5-2, 4-0 Northeast Conference) a 13-9 lead with 7:55 left in the third.
Esposito, who finished with 280 yards on 25-of-32 passing, threw a 5-yard TD to Andrew Smith early in the fourth quarter to give Albany a 20-9 lead.
The Great Danes controlled the ground game, rushing for 123 yards and holding Bryant to 30.
David McCarty led Albany's running effort with 118 yards on 38 carries.
The Bulldogs (3-3, 2-1) took a 9-6 lead when Vinton South caught a 13-yard touchdown pass from Mike Canfora early in the second half. The scoring pass was Canfora's only attempt in the game.
Michael Croce, who was sacked five times, was 22-for-35 with 234 yards for Bryant.