"Our guys don't give up," Lisch said. "We practice hard, we play hard. We never feel like we're out of the game. While there's time left on the clock you might as well play hard."
Saint Louis fell to 12-10 overall and 3-5 in conference games.
"I thought Xavier played a great game," SLU coach Rick Majerus said. That's the reason why they are No. whatever they are, 12 or 14. They are a terrific team. They've got great athletes. I thought our guys played with great heart and determination."
No. 13 Xavier improved to 19-4 overall and 7-1 in Atlantic 10 play.
Lisch scored his season-high 25 points on seven-of-11 shooting, including six of nine on 3-pointers, and made five free throws. He also added a game-high five assists and three steals.
"Kevin Lisch is one of my favorite players in the conference," Xavier coach Sean Miller said. "What he did down the stretch speaks volumes about his ability and shows how special a player he is. He hit tough shots. He almost single-handedly down the stretch beat us."
SLU shot 27.3 percent (six of 22) in the first half, 53.6 percent (15 of 28) in the second half and 42 percent (21 of 50) for the game. The Billikens made 10 3-pointers, including two from junior forward Barry Eberhardt.
Xavier led 13-12 early in the first half before SLU went on a 10-0 run to take a 13-12 lead on Eberhardt's first 3-pointer of the game and fifth of the season.
But then the Musketeers reeled off a 17-2 run to take control of the game and ran their lead up to as many as 16 points late in the first half.
Xavier led 35-22 at halftime.
The Billikens opened the second half with seven consecutive points — four from senior center Bryce Husak and a 3-pointer from Meyer — to cut the deficit to six.
"Saint Louis is an excellent defensive team," Miller said. "Seeing them up close, they have a method to their madness. They don't foul. They rebound, they're in terrific position, they have one of the best defenses we've faced. Their numbers speak for themselves. That's why when they're at home where they are more comfortable shooting, they are a really tough team to beat."
SLU trailed by six, 62-56, before Lisch knocked down a 3-pointer from the right wing with 1 minute and 1 second remaining. After Xavier's Brown scored inside, Lisch hit a 3 from the top of the key with 38 seconds left.
Xavier's Stanley Burrell then made two free throws before Lisch hit his six 3-pointer of the game from the top of the key with 27 seconds remaining.
After Xavier's B.J. Raymond made a pair of free throws, Lisch drove to the hoop and converted a conventional three-point play with 11.6 seconds left to tie the score at 68-68.
Xavier inbounded the ball and Lavender drove the lane and floated up a shot. Meyer came across to contest the shot attempt but Brown, a 6-foot-8 sophomore, was able to tip in the rebound with less than one second remaining to give the Musketeers the victory.