The Cavaliers trailed Illinois 58-56 when they that run began with 3 1/2 minutes left.
Beekman scored 17 points and was the tournament MVP. Kihei Clark and Jayden Gardner each added 12 for Virginia.
Jayden Epps scored 14 points for Illinois and Coleman Hawkins had 10.
The game was played exactly a week after Virginia football players Lavel Davis Jr., Devin Chandler and D'Sean Perry were fatally shot after a field trip to see a play in Washington. Two others were wounded. Former Virginia football player Christopher Darnell Jones Jr. faces three counts of second-degree murder and other charges.