The 27-year-old player was given a straight red card just five minutes into the match at Old Trafford for a tackle on Bryan Mbeumo.
Before his ejection, Manchester United scored twice, with Bruno Fernandes opening the scoring and Casemiro doubling the lead with a close-range header in the 37th minute.
Casemiro was also sent off just before halftime, but Chelsea was unable to capitalize on the opportunity despite a late goal from Trevoh Chalobah.
Sánchez took to Instagram to address the fans and his teammates, accepting responsibility for the team's loss.
In his message, he said, "On me today guys, big sorry for that, team put a massive effort on the second half but couldn't get there."