Xavier overcame a 38-32 halftime deficit by out-scoring BYU, 47-39, in the second half -- when Lavender scored 15 of his points.
Josh Duncan made a game-winning lay-up with 34.2 seconds remaining.
Justin Doellman had 23 points and five rebounds, and Derrick Brown had eight points and 16 rebounds for the ninth-seeded Musketeers (25-8), who have won four of their last five games.
Keena Young led BYU with 24 points, 10 rebounds and four assists. Mike Rose scored 15 points; while Austin Ainge added 12 points, three rebounds and three assists for the eighth-seeded Cougars (25-9), who lost their last two games of the season.