Rodgers, who shot 11 of 21 from the field, knocked down a team-high four 3-pointers and made 9 of 10 free-throw attempts.
Georgetown (2-0) trailed 30-28 at halftime but Rodgers opened the second half with two free throws and back-to-back layups to put the Hoyas ahead 34-30.
Delaware (1-1) rallied, cutting the deficit to 54-53 with 2:17 left to play behind a Lauren Carra 3-pointer, but Rodgers converted two free throws and recorded a steal and fast-break layup on the next possession to end the Blue Hens' comeback hopes.
Rodgers scored eight of Georgetown's last 10 points.
Delaware was without last year's leading scorer Elena Delle Donne. Delle Donne, who also missed Delaware's season opener, is dealing with symptoms of Lyme Disease. She is listed as day-to-day.