Barsy led by as many as 10 points in the first half, but a third-quarter comeback allowed Liepaja/Triobet to catch up and even take a 56-55 lead before the final period.
With three and a half minutes left in the game, Lauris Blaus had put the hosts up 70-63, but Barsy managed to make it a one-possession game after two free-throws by Edgaras Zelionis.
With 13 seconds left on the clock, Mikhail Yevstingneev tied things up at 76-76 with a jump shot, but Ingus Jakovics prevented an overtime by making one-of-two from the stripe on the other end after Donte Hill fouled him.
Ingus Jakovics had a team-high 18 points for Liepaja/Triobet, Renars Magone scored 17, as Lauris Blaus collected 16 points for the winners.
Barsy were led by Lithuanian big man Edgaras Zelionis, who had 20 points and eight rebounds in a losing effort, while his countryman Gytis Sirutavicius scored 18 points and grabbed six boards.