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The home supporters began to jeer when Sarri replaced Mateo Kovacic with Ross Barkley – the 20th time this season he has switched the two midfielders during a game.
There were also loud calls for the return of Chelsea legend Frank Lampard, while Manchester United fans jokingly chanted for Jose Mourinho, who was sacked from both sides.
First-half goals from Ander Herrera and Paul Pogba gave the visitors a solid lead as Chelsea once again struggled under Italian Sarri.
The former Napoli boss has come under increasing scrutiny for his apparent tactical inflexibility, having seen his side slip to fifth in the Premier League after a promising start.
The west London club have suffered three defeats from their last five league games and face an exit from the FA Cup.