Bedell's longest run of the day was for 39 yards and a score late in the first quarter, giving the Seawolves (4-4, 3-1 Colonial Athletic Association) a lead they would never surrender.
Conor Bednarski finished with 81 yards passing and a touchdown.
Bedell's TD run was the culmination of a 7-play, 75-yard Stony Brook drive following a 41-yard field goal by Elon's John Gallagher early in the first quarter.
Bednarski threw an 18-yard scoring pass to Adrian Coxson on the first drive of the second quarter to put the Seawolves up 14-3.
Gallagher scored the only points for the Phoenix (1-6, 0-3). He missed a second attempt from 36 yards late in the second quarter.
Elon QB Mike Quinn was 27-of-48 for 221 yards passing and threw three interceptions.