With the score tied 90-90, Hood was fouled by Vermont's Clancy Rugg. He missed the first foul shot but hit the second, providing the margin of difference.
The Catamounts couldn't get a shot off on their final possession, allowing the Blue Devils (5-1) to squeak out a fourth straight win.
Jabari Parker supplied 26 points with nine rebounds, while Hood added 22 points and seven boards for Duke.
Vermont (1-5) hung in until the final buzzer thanks to 24 points from Hector Harold and 20 from Rugg.
Celebrity Families of 2014 [PHOTOS]