A defensive error gifted the opening goal to Liverpool after only six minutes, but the hosts responded well to that early setback and had the better of things for much of the first half at St Mary's.
However, Liverpool added a second against the run of play just before half time, and Pellegrino admitted that his side's response in the second half was not good enough.
"We made it really difficult. After we conceded from a counter-attack it was a pity because we dominated the first half and played the best part of the game," he is quoted as saying by Sky Sports News.
"We created two or three clear chances but after their second goal we were down in confidence and belief and we couldn't react in the second half. I feel that this second half didn't show our reaction since January, because we were growing every single weekend.
"We didn't show what we are in the second half. The two goals were too much. We had to show another phase because we needed to score just one goal to be in the game until the end. The second half was really bad. Overall the anxiety was really high, we were not calm and gave the ball away all of the time."
The defeat leaves Southampton in the relegation zone and now one point adrift of safety.