Jose Mourinho challenged Chelsea to stick with him or sack him after the capitulation to Southampton.
Captain John Terry was restored but Saints exposed the now familiar failings in the defence which just five months ago formed the foundation for winning the title, with goals from Steven Davis, Sadio Mane and Graziano Pelle earning a first win at Stamford Bridge since January 2002.
The Blues, who had taken the lead through Willian's free-kick, were booed off after a fourth loss in eight Premier League games left them 16th in the table, one place below Bournemouth.
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In a seven-minute exchange with Sky Sports post-match, Mourinho said: "If the club want to sack me, they have to sack me because I'm not running away from my responsibility and my team.
"This is a crucial moment in the history of this club because if the club sacks me, they sack the best manager this club had and the message is bad results and the manager is guilty."
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He later added in his post-match media conference: "No way I resign. No way. Why? Because Chelsea cannot have a better manager than me.
"I want the best for my club and the best for my club is for me to stay."
Mourinho is in his second spell as Chelsea boss, his first from June 2004 was filled with success before unravelling in September 2007.
The 52-year-old, who was reappointed in June 2013 and signed a new four-year contract in August, insisted no alternative offer could persuade him to leave Stamford Bridge.
He insists Chelsea have learned from the managerial hirings and firings of Roman Abramovich's 12-year ownership and does not believe the Russian should speak out in support for the first time.
"The owner's profile all over these years of not speaking is fabulous. I would love to do the same," added Mourinho, who declined to say when he would next see Abramovich, who was at the match.
"The owner has that privilege. I don't think the owner needs a word, not even the club, the board, nobody. Not at all.
"When I was contacted to be back, I was told we had so many managers, we know you are the best.
"So I think it's time for the club to act in a different way, to mark a position, a position of stability, a position of trust.
"Three months ago we were all together in the bus with the cups. I think it's time for all of us to be together now in the bottom of the league."
The contrast between last term - when the Blues won the Capital One Cup and the Premier League title - and this is stark, but Mourinho maintains belief a top-four finish is probable.
He has almost conceded defeat in the title race, though, with Manchester City 10 points ahead.
"To be champions will be very difficult. Ten points (to) Man City, maybe tomorrow 11 points (to) Man United," Mourinho added.
"In football it's not impossible, but I think very difficult. Top four is completely - I don't say easy - but doable."
On the dissent from the supporters, Mourinho said: "If they have some negative reaction, I would do the same, because the results are not the results that we want."
But he maintains he has the support of his players.
"Do the players trust me and feel that I am the man for the job? This week we spoke about it and the answer was undoubted," he added.
Mourinho refused to analyse individual performances but denied humiliating Nemanja Matic, a half-time substitute who was hauled off 28 minutes later.
"I don't do that to anyone in football and in life," Mourinho said.
Mourinho suggested he would promote younger players, like Ruben Loftus-Cheek, after his senior players' struggles.
"Clearly it is easy to see that some players are in difficulties," he said.
"As a manager you have that situation of I like them, I trust them, I keep them, or I drop, I change.
"Maybe the situation is so negative that a young player feels less pressure than when the team is top of the league."
Chelsea in crisis overshadowed what was an impressive performance from Southampton.
"The whole team performance was perfect," Saints boss Ronald Koeman said.
"After 1-0 down we looked comfortable and of course, to win against teams like Chelsea you need your best players on that high level.
"Sadio was dangerous in his movements. It was impossible to have good defending against him."
Mourinho complained about Radamel Falcao being denied a penalty, even though Mane had a stronger claim.
It made little difference to Koeman.
"Maybe then it was 2-5 the final result, even with that," the Dutchman said.