Ugarte miskicked in front of goal in the 74th minute after Phil Barnes had pushed the ball straight into his path.
However, he was in the right place at the right time in the 87th minute when he brought the house down with a spectacular volley, which gave Barnes no chance and stopped the Gulls from taking three crucial points back south.
United had taken the lead in the 20th minute. Leon Constantine crossed from the left and Jo Kuffour gave Ben Foster no chance with a low shot.
Foster made two good saves during United's purple patch before Wrexham got on top, and it was Ugarte who held his head after his free kick was only just wide of a post.
It was continual one-way traffic during the second half, Chris Llewellyn dipping one shot over the bar and having another saved by Barnes.
It looked as though it was not going to be the home side's day when, in the 74th minute, Danny Williams' shot was parried out by Barnes who pushed the ball into the path of Ugarte, but the Spanish player miskicked when it looked easier to hit the ball into the gaping net.
After Ugarte had scored his eighth goal of the season, Torquay were under the cosh and survived the anxious last seven minutes - including stoppage time - before the final whistle sounded.