Galloway's shot will stir controversy. He used his right arm to push off Dayton guard Kyle Davis before knocking the shot down from the top of the arc with under 20 seconds to play.
Devin Oliver then missed a straight-away 3-pointer for Dayton at the other end and DeAndre Bembry made 1-of-2 free throws for Saint Joe's with one second on the clock.
Dayton inbounded the ball to midcourt after a timeout but couldn't get off a try with less than one second left.
Saint Joe's will face St. Bonaventure in the semifinals on Saturday after the Bonnies upset No. 18 St. Louis, the tournament's top seed, on Jordan Gathers' 3-pointer at the buzzer.
The fourth-seeded Hawks (22-9) lost their last two games of the regular season to George Washington and La Salle on the heels of an 8-1 stretch that put them in the conversation for an NCAA tournament bid and earned them votes in the AP poll.
They bolstered their chances with Friday's win over the fifth-seeded Flyers, who came into the tournament as the A-10's hottest team.
Dayton (23-10) opened the tournament with a 70-67 win over Fordham on Thursday and had fallen just once in its last 11 games, a 26-point loss at Saint Joe's on Feb. 25.
This one was close throughout, with neither team leading by more than six points. Saint Joe's took a 66-60 lead on a couple of Galloway free throws, which came after teammate Ronald Roberts shook the rim with a putback dunk off a missed 3-pointer.
Jordan Sibert, who scored 10, hit a 3-pointer from the right side after that to get Dayton within three. Matt Kavanaugh then made a layup and knocked down a baseline jumper to give the Flyers a 67-66 lead before Galloway's shot.
Kavanaugh had 10 points and Dyshawn Pierre scored 11 to lead Dayton. Halil Kanacevic had nine points and 12 rebounds for the Hawks and DeAndre Bembry scored 15.