Daryus Quarles added a season-high 15 points for the Hawks (18-10, 8-5 Atlantic 10 Conference), combining with Galloway to score 15 straight points in the first half and turn a one-point deficit into a 29-11 lead with 5:47 remaining.
Carl Jones scored 16 points and drained a 3-pointer with 1:50 left for a 67-60 lead after the Colonials pulled within four points.
The Hawks, who entered the weekend a half-game behind Xavier and Massachusetts for third place in the league standings, have won three straight and five of six.
Halil Kanacevic tallied 10 points and 13 rebounds as Saint Joseph's finished with a 46-30 rebounding advantage and 18 second-chance points.
"It wasn't a one-man show; it wasn't a guy kind of carrying us home on his back," said Saint Joseph's coach Phil Martelli following his team's fourth conference road win. "Across the board we got contributions,"
Tony Taylor scored 14 points and David Pellom had 12 points and 11 rebounds for the Colonials (9-17, 4-8).
Trailing by 17 points at halftime and 39-20 early in the second half, the Colonials rallied with runs of 10-0 and 6-0. Taylor's layup following a Hawks turnover pulled George Washington closer at 58-54 with over five minutes left, but the Colonials were unable to finish the comeback.
George Washington has lost six of seven and four straight to the Hawks on their home court.
"We weren't able to give effort for 40 minutes," said Colonials coach Mike Lonergan. "Just a huge hole to dig (out of) against a pretty good team."
Six points into the early run, Galloway drained consecutive 3-pointers and tacked on a driving layup as the Hawks led 22-9. Quarles, who entered the game averaging fewer than five points per game, followed a Pellom basket with his own 7-0 run for a 29-11 lead.