Denis Smith's side were grateful for Chesterfield's win over Oldham - a result which keeps the Welshmen's survival dream alive.
With plenty at stake for the visitors, they managed to get off to an unbelievable start with four goals inside the first 29 minutes.
Carlos Edwards played a neat through ball to Ugarte and the Spanish hit-man rifled home a superb angled drive on five minutes.
With the visiting fans still celebrating, Ugarte then fired home his second, albeit via a deflection, following a good cross from the left from Andy Holt.
An amazing opening to the game continued when Edwards was allowed the time and space to tear through the Stockport midfield before rifling home a smart low drive.
There's no doubt struggling Stockport were stunned, but they got themselves a goal on 27 minutes.
Adam Le Fondre fired in a low cross across the face of goal and there was veteran Stuart Barlow who bundled the ball home from all of four yards.
But it didn't take long for Wrexham to net again - just two minutes.
Holt made the most of some poor defending and flicked the ball to Ugarte who nodded home to complete a wonderful first-half hat-trick.
To their credit Stockport kept battling away in their final home game of a miserable season and before the break Damien Allen and Michael Raynes both came close to a second goal.
After the break Stockport continued to strive away for an opening and Le Fondre should have done better but he fired a shot straight at goalkeeper Ben Foster.
At the other end home goalkeeper James Spencer had to be at his acrobatic best to beat out a powerful Holt volley.