Evan Conti added a career-high 22 points for the Bobcats (3-2), who led 43-24 at halftime.
Hector Harold and Ryan Pierson had 13 points apiece for the Catamounts (3-3).
Quinnipiac shot 55 percent (28-51) and went 28 of 32 from the foul line with Drame making 13 of 16. Vermont shot 45 percent (23-51) and went 24 of 30 from the line but got crushed on the boards. The Bobcats had 40 rebounds and their 17 offensive boards were one more than Vermont's total of 16.
Leading 9-6 less than 5 minutes in, Quinnipiac scored eight straight to get a double-digit lead and after four Vermont points, the Bobcats scored nine straight. Conti had nine in the 17-4 run to make it 26-10.