The Red Devils got back to winning ways after a Romelu Lukaku double gave Jose Mourinhos side all three points against the Clarets.
Sanchez was the architect for Lukakus opener as he lofted a fine ball to the back post for the Belgian to nod past Joe Hart.
The Chilean, though, was removed shortly after the hour mark and appeared far from pleased with the decision as he threw a tantrum on the bench.
Sanchez could be seen throwing his taping on the ground and sat alone on Uniteds bench, cutting an angry figure.
However, the 30-year-old appeared to have calmed down by the end of the game and sent out a celebratory message on Instagram after his best performance of the season.
Sanchez simply put: Come on United in a message that he also shared to his 2.7m followers on Twitter.
There is now a 12-day international break before the Premier League resumes when United take on unbeaten Watford at Vicarage Road.