Jabarie Hinds added 17 points for the Minutemen (7-5), who made 21 of 27 free throws, including 10 of 12 by Davis. Seth Berger grabbed 10 rebounds to lead a 45-38 advantage on the boards.
Jerome Frink led the Blackbirds (6-5) with 21 points and 13 rebounds, his second double-double this season. Martin Hermannsson added 18 points and Joel Hernandez 13.
UMass took the lead for good at 70-66 with 3:35 left behind eight straight points from Davis.
Hernandez hit a 3-pointer with 10 seconds left to get LIU within two but Davis made two free throws and LIU missed a 3-pointer and a free throw in the final seconds.
The Minutemen have won two of their last three games after a three-game losing streak.