The Miners led by five points with two minutes left, but Matt Stainbrook and Myles Davis both made jumpers for Xavier to cut the difference to two points with 19 seconds to go.
Trailing 76-73 with three seconds left, the Musketeers turned over the ball on the inbounds play to send UTEP to the line. The Miners shot 4 of 6 on free throws in the final 10 seconds to stay ahead.
Hunter scored 17 points, Earvin Morris added 15, and Cedrick Lang finished with 13 points and 10 rebounds for the Miners (4-0).
Davis led Xavier (5-1) with 15 points, and three other Musketeers finished in double figures.
UTEP will play in the Wooden Legacy championship on Saturday.