Anthony Martial equalized 10 minutes after the break when the Chelsea keeper kept out a close-range shot by Juan Mata.
It all took place after Barkley made it 2-2 in the 96th minute to earn the draw.
Mourinho's touchline demeanour changed as he hugged Martial when the France forward was replaced in the closing stages by Andreas Pereira.
He said: "I'm not annoyed with anything".
Afterwards, Mourinho told Sky Sports that Sarri's assistant had apologised to him and he had accepted.
All eyes will be on Stamford Bridge on Saturday when Jose Mourinho returns to face the club he managed twice in his career.
His collected demeanour did not hold up for long though, as he held up his three fingers to signify the number of Premier League titles he won in the direction of the home supporters, who were baiting him as he made his way off the pitch.
"I did not get respect from the Chelsea fans", he said.More news: G League to offer 'select contracts' to elite prospects
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"I have a certain level of education socially and in sport and what I did today, I would do in Porto, Madrid, Milan".
He had spent two years as technical assistant to Sarri in Italy before becoming his second assistant coach at Stamford Bridge. "I have seen the match on the bench, and now want to study the second half of the match".
I spoke with Jose and I spoke with my staff because we were on the wrong side. "I have dealt with the situation immediately", the Italian said. "I will speak to him again tomorrow".
Mourinho had to be restrained by stewards before Chelsea manager Maurizio Sarri managed to reconcile the Manchester United manager.
Mourinho, meanwhile, is embattled at United, who are eighth after three losses in the opening eight games.
Mourinho demonstrated he can still provoke a reaction from these United players as they responded to his half-time talk with a transformed second-half display, full of pace, threat and intent, which took them to within seconds of an outstanding win.
Police later disputed Mourinho's explanation an escort was refused for the team bus journey from a city center hotel.
Paul Pogba was at fault for Chelsea's opener, leaving Antonio Rudiger in acres of space to head home from Willian's first-half corner but Mourinho refused to criticise his midfielder. Everton look to consolidate their improvement under Marco Silva at home to Crystal Palace, who need to recover some of the form they showed at the start of the season in order to calm a few nerves.