The 27-year-old was shown a straight red card by referee Peter Bankes after a tackle on Man United forward Bryan Mbeumo.
Sanchez had appeared uncertain in the early stages of the game and his decision to leave his goal area to challenge Mbeumo backfired.
As a result, Sanchez became the earliest Chelsea player in Premier League history to receive a red card.
Manchester United's Bruno Fernandes scored the opening goal nine minutes after Sanchez's red card, and Casemiro doubled their lead on the 37th minute.
Casemiro was also given a red card before halftime, leveling the playing field.
Despite a late goal from Trevoh Chalobah, Manchester United held on to win the match.
