Washington scored nine points during a 25-4 run that gave the Gaels a 19-point lead midway through the first half.
Drexel (2-9) used a 12-2 spurt that spanned halftime to pull within eight on a 3-pointer by Sammy Mojica with 17 minutes to play, but Iona scored 15 of the next 19 to make it 58-39 with 11:04 left.
A pair of free throws by Tavon Allen capped a 13-3 Drexel run and cut the Dragons' deficit to 72-70 with 1:38 remaining, but Iona made 5 of 8 free throws from there to seal it.
Drexel missed its last field goal attempts.
Mojica led the Dragons with a career-high tying 20 points.