Shaun Noriega added 15 points for the Bulls. Augustus Gilchrist and Victor Rudd Jr. each had 12.
Four McGlynn led Vermont with 17 points, but he missed a game-tying layup with 1 second left.
Matt Glass finished with 13 points, but his put-back of the final rebound reached the rim a split-second too late.
Luke Apfeld had 10 points for the Catamounts, who finished last season with five consecutive road victories.
Blake Nash hit a 3-pointer midway through the second half to give South Florida its largest lead at 49-37.
Apfeld capped a Vermont rally with a layup with 10.5 seconds to go, trimming the deficit to 60-59. Noriega was immediately fouled and hit the second of two free throws at the other end.
There were four lead changes before South Florida took a 31-27 halftime lead with eight points apiece from Anderson, Noriega and Rudd.
McGlynn hit 3 of 4 from long range and had 11 points at the break. The Catamounts outshot the Bulls from the field in the first half but were outscored 7-0 at the free-throw line.