Piatkowski's 35-yard pass to Paddock tied the game at 13-13 in the second quarter for the Bulldogs (5-1, 3-0 Pioneer Football League).
After Drake blocked an Eagles' field goal try, the Bulldogs moved 54 yards in nine plays to go ahead 16-13 on Billy Janssen's 33-yard field goal 10 seconds before halftime.
In the third quarter, Piatkowski hit Paddock for an 80-yard touchdown with 6:35 left and found Joey Orlando for a 15-yard score on the final play of the quarter to make it 32-20. Piatkowski completed 16 of 27 for 310 yards with one interception.
Paddock had five catches for 163 yards and Janssen kicked four field goals.
Morehead State (2-4, 1-2), which came in leading the Football Championship Subdivision with 551.2 yards per game, was held to 297.