The Saints conceded first-half goals to Roberto Firmino and Mohamed Salah at St Mary's to drop into the bottom three of the Premier League, a point adrift of Huddersfield Town who were big winners earlier in the day.
Under-fire boss Pellegrino understands Southampton supporters' frustrations, but insists that the defeat to Liverpool did not provide a true reflection of their recent results, having gone six games without defeat in all competitions prior to Sunday.
"We are in one situation, but we have to keep going, keep fighting," he told Sky Sports News. "I feel that this second half didn't show our reaction since January, because we were growing every single weekend.
"I am convinced we have got good pace to try to win the next game. It will be long until the end, we have to be there, recover our mentality and in football everything can change in two or three weeks."
Southampton have won just once in their past 14 Premier League games, keeping only one clean sheet in the process.