West Ham now have an 11-day break before visiting Crystal Palace on Boxing Day.

Asked if he will be in charge at Selhurst Park, Pellegrini replied: "You never know, you never know.

"I don't think that your job depends on whether you win or you lose, I think that it depends on the trust of the owners of the club.

"And I'm absolutely sure that they trust in what we do but I cannot answer that question."

Speaking about pressure, he continued: "It is a question that I always answer the same way.

"For me it is exactly the same the week after we beat Manchester United (and) after we lost against Arsenal. Exactly the same.

"We need to win the next game, you must try to play better.

"Winning or losing you must analyse mistakes and what you did well and I don't think that just because you are losing you must feel pressure.

"In this profession, you must feel pressure always."

West Ham were the dominant force on the south coast and deservedly went ahead when £45million summer signing Haller scuffed home his first goal since early October in the 37th minute.

The Hammers had earlier seen a strong penalty appeal waved away by referee Martin Atkinson, before Antonio's thunderous 52nd-minute strike was ruled out for a handball in the build-up following a review.

Haller and Antonio caused constant problems for the hosts and Pellegrini was impressed with the link-up play between his lively attacking duo.

"I'm very happy for him (Haller) because I think he returned from the first game in the Premier League, maybe after that for different reasons he was not doing his best performance," said Pellegrini.

"I think playing with Antonio works better for him.

"Antonio had a lot of pace and created space and movement. (I am happy with Haller) not only for scoring but the other chances he had and the way he worked for the team.

"From the first minute of the game we tried to win.

"I like this personality of the team and I think that is because the players believe in what we do."

Southampton remain inside the relegation zone on goal difference after failing to build on successive home wins over fellow strugglers Watford and Norwich.

Danny Ings, bidding to score for the sixth consecutive match, struck the crossbar 20 minutes from time but the hosts paid the price for a dismal opening 45 minutes.

Saints boss Ralph Hasenhuttl says his team will continue to fight for survival.

"We know that the relegation goes until May and like last season it's about battling for staying in (the division), everybody knows that," said the Austrian.

"And we know the situation, we know what it's about to stay in the league and we know that it's a tough fight until the end and we will take this fight.

"Everyone has pressure and my job is to look on the game and on (things) separate from the result.

"Then I must find the right decisions for the next week because it is the only way to get back in the successful track."