The Southampton boss will come under intense scrutiny after the 2-1 defeat saw his side fall one place above the relegation zone.
Shane Long scored his first goal in 11 months to hand the Saints the lead.
However, despite dominating the first-half, goals from James McCarthur and Luka Milivojevic won the game for Palace.
Pellegrino said: "It is a difficult game for us to digest because in the first half it was completely different and we were talking about how to score the second goal, but we gave them the possibility to score a couple of goals because we played backwards and sometimes we played too deep.
"My player's confidence will be down but that is normal. We must learn how to play this type of game, especially when the conditions are like that.
"I think in the second half our goal was to keep the ball and try to keep possession in the opposition half, but the concern for me now is lack of confidence.
"Pressure is part of our job, I understand people are disappointed like us but when one side are winners the other must be losers."