The dramatic victory lifted the 10th-seeded Hawks into a quarterfinal meeting on Friday with 22nd-ranked Virginia Commonwealth.
Charlotte, Butler and Massachusetts also advanced to the quarterfinals -- Charlotte doing so in controversial fashion.
Xavier had an eight-point lead midway through the second half, but the Hawks caught up with 8:30 to go and it was a close game the rest of the way.
Semaj Chirston made two free throws to give Xavier a 57-56 lead with 1:12 remaining and neither side could score again until Chirston fouled Galloway just before the buzzer.
The two free throws by Galloway left the Hawks with a 19-12 record. Galloway led St. Joseph's with 14 points while Travis Taylor had 16 for Xavier.
Charlotte earned a quarterfinal chance against top-seeded St. Louis with its 68-63 win over Richmond in a game that involved a bizarre finish.
Richmond was called for three technical fouls in the last 4.7 seconds, leading to eight consecutive free throws by Pierria Henry that turned a three-shot deficit into a 68-63 victory for Charlotte.
Henry finished with 28 points and 12 rebounds in boosting the 49ers to 21-10.
Butler (25-7) defeated Dayton 73-67 and will take on LaSalle in the quarterfinals. Butler won its third straight game thanks in part to 18 points from Andrew Smith.
Massachusetts ended the first round by scoring the final six points in a 77-72 decision over George Washington.
The Minutemen (20-10) took the lead for good on a jumper by Chaz Williams with 1:02 to play and George Washington suffered two turnovers and two missed shots the rest of the way.
Massachusetts will play the last of the quarterfinal games on Friday against Temple.
Senate Democrats to pull all-nighter on climate change
Teacher apologizes for showing sexual image of herself in class