Fairfield (3-3) led through much of the extra period before Columbia's Alex Rosenberg hit a 3-pointer and Luke Petrasek's dunk briefly gave the Lions an 81-80 lead with 26 seconds to play. A foul on Columbia's Maodo Lo with 12 seconds left set up Nelson's winning trip to the free throw line. Lo missed a 3-pointer with 3 seconds to play.
Columbia (3-3) had forced the extra period with a 3-pointer from Grant Mullins to tie it at 72-all with 6 seconds left in regulation.
Nelson shot just 3 of 16 from the field but sank all four of his free throw attempts to finish with 11 points. Matija Milin added 10 points to go with 11 rebounds.
Lo finished with 23 points to lead four Columbia players in double figures.