Ryan Anderson added 12 points for the Eagles (7-10, 2-1 Atlantic Coast Conference). Boston College beat Clemson, 59-57, for its first league win on Thursday.
Virginia Tech's leading scorer Erick Green was sidelined with a sprained left knee. Virginia Tech shot just 36.2 percent (17 of 47).
The Hokies lost their first two conference games by three points to Wake Forest and four to Florida State.
The Eagles had opened an 11-point lead with 3:40 left before Robert Brown nailed consecutive 3s to trigger a brief spurt that closed it to 58-53 with 2:16 left. But Virginia Tech rushed some quick shots and BC turned the ball over before Gabe Moton's one free throw gave the Eagles a six-point edge with 26 seconds left
The Hokies did cut it to the final margin on Jarell Eddie's 3 with three-tenths of a second, but the Eagles inbounded the ball of Clifford's hands and the buzzer sounded as it was in the air before a Hokies' player could grab it.
Boston College led 44-40 midway into the second half before scoring 12 of 17 points over a 6:21 stretch to take control. The first eight points came from the free throw line before Humphrey banked in a driving one-hander and Anderson nailed a one-handed flip in the lane after grabbing an offensive rebound, giving the Eagles a 56-45 advantage with 4:34 left.
The Eagles took a 29-27 lead into the locker room at halftime thanks to a quick offensive outburst by point guard Jordan Daniels. He nailed a 3 from the left corner with 32 seconds left and hit a one-handed runner from the edge of the lane as the buzzer sounded.
In a second half filled with whistles and shoddy offense, the Hokies were held without a field goal for the initial 8:01, but they stayed close by hitting 9 of 10 from the foul line.
The Eagles had opened a six-point edge four times in the opening nine minutes after halftime before they pushed it to double digits (52-42) on Anderson's two free throws with 7:33 to play.
Green injured the knee in practice Thursday and was a game-time decision. He was on the bench wearing a sweat suit.
Green led the Hokies with 21 points in their last game -- a 63-59 setback at home against Florida State. Green is averaging 16.1 points per game and has scored in double digits in 18 straight, the league's longest active streak.
BC guard John Cahill left the game with a right shoulder injury early in the opening half and was on the bench with it wrapped the rest of the game.