The Saints saw a two-goal lead slip away at St Mary's Stadium on Saturday afternoon, requiring a disputed penalty three minutes into added time to come out on top.
After seeing his side end their six-game goal drought on home soil, recently-appointed Pellegrino admitted following the match that remaining alert for 90 minutes is the next challenge for his group of players.
"Maybe [West Ham] are stronger than us and they were playing this type of game with free kicks, long free kicks, deep free kicks," he told reporters. "The majority of the problems for us were in this type of moment.
"After the second goal we were a little bit relaxed but not in terms of concentration, in terms of trying to be more clinical to score the third one and try to kill the game. When you give the opponent some opportunities like today you will be in trouble and this is something we have to improve."
More goals were scored in Saturday's match on the south coast than in the Saints' previous five home matches combined (three).