The 27-year-old was picked up a straight red card just five minutes into the game at Old Trafford for a rash tackle on Bryan Mbeumo.
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Bruno Fernandes opened the scoring shortly after before Casemiro doubled their lead in the 37th minute with a close-range header.
Casemiro was then sent off just before half time, but despite a late Trevoh Chalobah goal, Chelsea were unable to capitalise.
Sanchez has since taken to Instagram to send a message to the fans and his teammates, accepting responsibility for the defeat.
He said: “On me today guys, big sorry for that, team put a massive effort on the second half but couldn't get there.”