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.
South Dakota (0-5) was paced by Larson with 23 points and 12 rebounds. Bos added 20 points.