Roberts had 13 points and 11 rebounds for the Hawks (9-5, 1-1), who have won four of their last five games.
Saint Joseph's led by 11 early, but the Dukes (7-9, 0-2) fought back to tie it at 51 with 9:35 left on Quevyn Winters' 3-pointer. Roberts answered with a three-point play, then Langston Galloway, who led the Hawks with 16 points, added two free throws.
After Duquesne got to 56-55, Papa Ndao and Galloway hit back-to-back 3-pointers.
The Dukes kept threatening, closing to within two points twice, the last time at 66-64 with 2:45 to go. Galloway's back-to-back jumpers and the Hawks' 4-of-8 shooting from the free-throw line sealed it.
Andre Marhold had 13 points and 11 boards for Duquesne, which has lost four straight.