Spurs looked set to claim a vital win in their quest for Champions League qualification when Ivan Perisic fired home in the 74th-minute after earlier efforts by Pedro Porro and Harry Kane.
Bottom-of-the-table Southampton instead mounted a fine fightback with Che Adams and Theo Walcott reducing the deficit before James Ward-Prowse fired home a stoppage-time penalty to extend Tottenham's winless run on the road to five matches.
Recent defeats at AC Milan and Sheffield United saw Spurs exit the Champions League and FA Cup to make it a 15th consecutive season without silverware and their manager took aim at all parties in an astonishing 10-minute interview following their collapse at St Mary's.
"If we are going to discuss the penalty, it means we don't want to see the other situations," Conte said.
"I think it's much better to go into the problem because the problem is that, for another time we showed that we are not a team.
"We are 11 players that go into the pitch. I see selfish players, I see players that don't want to help each other and don't put their heart (in).
"Why? Because they are used to it here, they are used to it. They don't play for something important yeah. They don't want to play under pressure, they don't want to play under stress.
"It is easy in this way. Tottenham's story is this. Twenty years there is the owner and they never won something but why? The fault is only for the club, or for every manager that stays here. I have seen the managers that Tottenham had on the bench.
"You risk to disrupt the figure of the manager and to protect the other situation in every moment.
"Until now I try to hide the situation but now, no because I repeat, I don't want to see what I have seen today because this is unacceptable and also unacceptable for the fans.
"They follow us, pay for their ticket and to see the team another time, to have this type of performance is unacceptable. We have to think a lot about this."
Last weekend's victory over Nottingham Forest had lifted some of the gloom around Spurs but uncertainty over the managerial position remained with Conte's contract set to expire in the summer and no talks over a new deal under way.
Quizzed on if the lack of clarity over his future had affected the squad, the 53-year-old pushed back at another "alibi" for his under-performing players during a passionate monologue.
He added: "The club has the responsibility for the transfer market. Every coach that stayed here has the responsibility and the players? The players? Where are the players?
"In my experience, I can tell you that if you want to be competitive, if you want to fight, you have to improve this aspect and this aspect, I can tell you, in this moment is really, really low.
"And I see only 11 players that play for themselves.
"There are 10 games to go and some people think we can fight. Fight for what with this spirit, this attitude, this commitment? What? For seventh, eighth, 10th place? I am not used to this position.
"I'm really upset and everyone has to take their responsibility. Not only the club, the manager and the staff, the players have to be involved in this situation because it is time to change this situation if Tottenham want to change.
"If they want to continue in this way, they can change the manager, a lot of managers, but the situation cannot change. Believe me."
While Conte threatened to talk himself out of a job, Saints rookie boss Ruben Selles reflected on a well-deserved point for his side.
Southampton remain bottom of the table but are only two points off safety and Adams' second-half goal made him the first striker to score for the club in the league in 2023.
Selles: "We didn't deserve to be 3-1 down. I think we put a good performance in the pitch and it was simply too much.
"The game can go in another direction but you can see the togetherness from the captain to the rest of the team.
"Together with the crowd, we made an impact. We believed we could score. We made the second and then we got the third one. It was an absolute pleasure to be there."