Goals from Ander Herrera and Paul Pogba earned Ole Gunnar Solskjaer a sixth successive away win and a place in the FA Cup quarter-finals, against Wolves.
Sarri was subjected to jeers, boos and chants to be left in no doubt the strength of the Chelsea fans' feelings towards him, just seven months since his appointment.
Chelsea fans joined in as United supporters sang "you're getting sacked in the morning", sarcastically applauded predictable substitutions and shouted "f*** Sarri-ball" in opposition to the possession-based tactics the 60-year-old boss favours.
Asked if he had heard that chanted at him before, Sarri said: "No. For everything, there is a first time."
The Italian added: "I am worried about the results. Not about the fans.
"I can understand the situation and our fans, because the result wasn't really good. We are out of the FA Cup, so I can understand. But I am worried about our results."
Chelsea have little time to dwell on the defeat, with Thursday's Europa League last-32 second leg with Malmo followed by Sunday's Carabao Cup final with Manchester City and next Wednesday's Premier League clash with Tottenham.
Whether Sarri sees out the run of fixtures remains to be seen.
City inflicted a 6-0 loss on Chelsea in their most recent Premier League game, the Blues' worst defeat in 28 years, and further defeats may see Sarri's reign end. The Italian clung to the positives.
"If we are able to win three or four matches in a row, it will be really very easy [to change the negative mood]," Sarri added.
"I was really very worried when I was in League Two in Italy. Not now.
"I think the players are with me. Of course I am not sure, but I think so. The relationship is not so important. It's important to play, it's important to get good results, but my relationship with my players is good at the moment."
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