The final whistle was marked by ugly scenes around the Stamford Bridge tunnel after one of Sarri's assistants, Marco Ianni, riled United boss Jose Mourinho by celebrating Barkley's equaliser in front of the visiting dugout.
Yet Mourinho ended up questioning why Dean played so much stoppage time after Barkley snatched a point for the hosts in the 96th minute.
"I did not get respect back from Chelsea fans but that is not my responsibility", Mourinho said.
"I'm not annoyed with anything".
He told BBC Sport: "I didn't see what happened but I have spoken to Jose Mourinho". What I did here today I will do in Madrid, in Milan, in Porto, the reaction from the fans is not up to me. I have made a lot of mistakes in my career, especially when I was young like him. They didn't see it coming.
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"I am very happy with the performance for 60 minutes because for 60 minutes we played our football", the Italian coach said. But two second half goals from Martial helped United to one of the shock results of the season so far. "I want to play with short passes, stop".
The result preserves Chelsea's unbeaten start to the Premier League season, but Sarri admitted his team were fortunate to escape defeat after resorting to a more direct style of play following Martial's equaliser.
And there was enough in this game to anger and delight him - a awful opening goal, when Pogba lost his man, a stirring fightback, a performance which highlighted Martial's brilliant potential and then the gut-wrenching Barkley equaliser, just as the points seemed to be in the bag.
Chelsea's slick passing game gave way to long-ball hoofing tactics while José Mourinho's United stepped up a gear and Martial scored again on the break in the 74th. We conceded a goal from a set-piece, but we were in the game so the distance between the first and the second half was not as big as it was against Newcastle.
"After that his assistant came to me in Sarri's office and apologised to me". "Normally when I am losing I never get six minutes". But I have that situation immediately [after the match].
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