Quick completed 10 of 18 passes for 94 yards and also rushed for 82 yards for the Aggies (2-1, 1-1). Cohen carried 23 times for 96 yards.
Quick connected with Denzel Keyes from 12 yards for the game's first score with 1:27 left before the break and the Aggies led 7-0.
After missing two field goal attempts in the first half, Elon tied it up 7-7 on a 16-yard touchdown run by quarterback Connor Christiansen early in the fourth quarter.
The Aggies broke into the lead again with Cohen's 13-yard touchdown run with 5:22 to play and held on for the win.
Christiansen finished with 143 yards on 19-of-32 passing for the Phoenix (1-2, 0-1). The Aggies held Elon to just 88 yards rushing.
AP college football website: collegefootball.ap.org