Nelson's basket with 4:34 left in regulation gave Fairfield a 68-57 lead, but the Stags went scoreless the rest of the way. Tyler Larson scored six of South Dakota's final 11 points, including a game-tying layup with 40 seconds left.
In overtime, Fairfield made 8 of 8 free throws and its defense held South Dakota to just two Brandon Bos field goals.
Mike Kirkland Jr. added 20 points, 10 coming from the free throw line, for Fairfield (2-3). Amadou Sidibe had 14 points and 12 rebounds, and Marcus Gilbert chipped in with 12 points.
South Dakota (0-5) was paced by Larson with 23 points and 12 rebounds. Bos added 20 points.