Bembry scored 16 of his 22 in the second half, Miles made 3 of 7 from 3-point range and Aaron Brown chipped in 13 points as the Hawks (2-2) bounced back from Wednesday's 52-point loss at No. 13 Gonzaga.
Saint Joseph's overcame a slow start due to an early shooting slump and controlled the second half, taking its largest lead at 46-32 with 13:30 to go.
Vermont (2-2) pulled within 61-56 on Ethan O'Day's layup with 3:23 left. The Hawks responded with five straight points and held on from there.
The Catamounts opened the game on an 11-1 run before the Hawks took the lead for good during an 11-0 run to go up 19-16.
O'Day and Hector Harrold scored 10 apiece for Vermont.