"We didn't want the season to end," Galloway said.
It almost did. Xavier (17-14) missed converting a length-of-the-court pass when Isaiah Philmore missed a layup at the buzzer.
The Hawks fought off a 10-0 run that left them down by eight points with 12 minutes, 30 seconds left in the second half.
Trailing by 41-33, St. Joseph's suddenly showed fire and a renewed sense of purpose.
The Hawks stormed back, taking the lead at 45-44 on a three-pointer by Halil Kanacevic with 8:40 left. From there it remained close.
And then came the final 1.4 seconds.
Galloway was fouled driving to the basket by Semaj Christon, the A-10's rookie of the year.
"It was a legit foul, and I have no complaint; it's the right call," Xavier coach Chris Mack said. "I wish we had the toughness to step in and take a charge."
The first free throw wasn't the smoothest by Galloway. It rattled the rim but finally went in, tying the score at 57. After a Xavier timeout in an attempt to ice Galloway, he hit nothing but net for a 58-57 lead.
Xavier needed a miracle, having to inbound under the basket. Then again, divine intervention could have played into the Hawks' fortunes.
"With two Jesuit schools, I did leave a ticket for Pope Francis," St. Joseph's coach Phil Martelli quipped. "I wish him well."
On the final play, Xavier's Justin Martin, threw a long inbounds pass that bounced off the backboard right into the hands of Philmore, who missed his chance. The Hawks had escaped.
Galloway led the Hawks with 14 points, but equally important was the defensive job he did on Christon, who had 10 points but shot 3 for 14.
Contact Marc Narducci at firstname.lastname@example.org. Follow @sjnard on Twitter.