The late strike marked the third time Everton have conceded in stoppage time this season.
It only took nine minutes for Viktor Gyokeres to put the Gunners ahead.
The Blues’ poor run also included defeats to Paris Saint-Germain in the Champions League.
Beto doubled the lead for Everton for the hosts in the 62nd minute, firing home a shot from close range that slipped in between Sanchez’s legs.
The Gunners secured a crucial win towards the end of the game, scoring their two goals in the 89th and 97th minutes.
Tomas Soucek pulled one back for West Ham in the 49th minute when he guided home Malick Diouf’s pass from close range.
United remain fourth on the table, with one point ahead of Chelsea, which was forced to a 2-2 draw at Stamford Bridge by Leeds United.
David Moyes’ side remain in the top half and will look to build on that after Christmas when they travel to Burnley.
Everton beat Manchester United 1–0 as David Moyes’ 10 men secured a huge three points, handing the manager his first away victory at Old Trafford on his 18th attempt.
Chelsea have now gone four points adrift of Liverpool, who can extend the lead to seven points if they beat Leicester City in their game later on Thursday.
