A 3-pointer by McDougald broke the game's 11th tie, giving the visiting Dolphins (10-10, 5-2) a 59-56 edge with 8 minutes left.
The Ospreys (8-12, 3-4) cut the lead to one point four times after that, getting to within 69-68 with 2:37 to play.
Glenn Powell, who scored 14 points for Jacksonville, responded with a jumper, and the Dolphins scored the game's final six points from the free-throw line.
Both teams shot well in the second half. Jacksonville made 60.9 percent (14 of 23) from the field and North Florida made 57.1 percent (16 of 28).
Parker Smith scored 19 points for the Ospreys, including 4 of 9 3-pointers. That matched his season average of 4.0 treys per game, tied for first in the nation.
South Carolina will enter next season with Sindarius Thornwell, Duane Notice, P.J. Dozier and a lot of questions following a turbulent offseason.
Ben Simmons explains what he is focusing on ahead of the start of his NBA career.
After John Calipari's comments about the Louisville scandal, Rick Pitino says that Calipari should not be worried about other programs and to keep quiet unless he is praising another institution.