Sullivan scored eight points, including two 3-pointers, during a 15-3 run that gave the Bears (1-0) a 46-38 lead. McGonagill's 3-pointer made it 51-45 with 7:12 remaining after the Bearcats (0-2) cut the lead to three points.
Brown held Binghamton to one basket the rest of the way to win in Mike Martin's debut as Bears coach.
Freshman Cedric Kuakumensah grabbed a game-high nine rebounds for the Bears, while sophomore Rafael Maia, a Brazilian native who was ineligible for his freshman year because of an NCAA rule, scored seven points, including six in the second half.
Maryland point guard Melo Trimble, considered a lock to leave the program before struggling in the second half last season, has withdrawn from the NBA draft and will return for his junior season.
Louisville's Chinanu Onuaku, who recently underwent a medical procedure to correct a heart rhythm issue detected during the combine, will keep his name in the NBA draft pool.
Andrew White III will return to the Nebraska men's basketball team next season after removing his name from the NBA draft.