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.
Maria Taylor and Peter Burns share the greatest moments of the year around the conference.
SEC Network looks back at all of the NCAA Championships won by the SEC this year.
SEC Network's Summer Tour stops by Baton Rouge to visit with Les Miles and the Tigers.
Three-time NFL MVP quarterback Brett Farve shares his stories of coaching high school football at the LSU Coaches Clinic.
SEC Network looks back at the highs and lows of Tigers' season.
Baylor has released five signees from its 17th-ranked recruiting class, including J.P. Urquidez, Kameron Martin, Donovan Duvernay and Patrick Hudson. Sources said the others was Parrish Cobb.