Cohen, who finished with a career-best 234 yards on 20 carries, threw a 2-yard pass to Denzel Keyes to give the Aggies a 10-3 lead in the first quarter.
John Gallagher kept Elon (1-1) in the game by kicking four field goals.
Elon held a 12-10 lead early in the final quarter when Cohen broke free for the winning score with 11:40 remaining.
Elon was driving in the Aggies' territory when defensive lineman Angelo Keyes reached up at the line of scrimmage and grabbed Mike Quinn's pass at the NC A&T 34. Tony McCrae broke up 5 passes and finished with nine tackles. Landis Shoffner had 18 tackles for NC A&T (2-1).