The Saints dominated the game from the off and could have had a penalty early on when Nemanja Vidic handled in the area, but referee Mike Dean waved play on.
Luke Shaw had a good chance to hand his side the lead when he played a one-two with Adam Lallana and the midfielder knocked a cross into the box for the 18-year-old, but his header went over the bar.
Southampton did take the lead in the 28th minute when Steven Davis released Rickie Lambert one-on-one with David de Gea, and the England international made no mistake when slotting the ball past the Spaniard.
After the break, United responded almost immediately and found an equaliser as Mata curled a free kick from the edge of the area into the top corner past Artur Boruc.
The Red Devils could have taken the lead when Adnan Januzaj raided down the left and curled a cross into the middle for Danny Welbeck, but he headed his close-range effort over the bar.
United pressed forward again and Fletcher joined Welbeck in missing an excellent opportunity as he could only head wide from Chris Smalling's cross into the box for what would have been the Scot's first goal of the season.
Both sides pressed for a winner, but were forced to settle for a point apiece on the final day.