Like their rivals, Everton have also struggled, with four defeats in six games since caretaker David Unsworth took over from the sacked Ronald Koeman, and Pellegrino is aware a first victory in four games will ease the pressure on his team.
"It will be important, because we can change our face, increase our confidence in our job, our players, our beliefs, and the people who work around the team," Pellegrino said.
"Everybody needs a good game, but the most important thing is trying to play the game that we deserve to win.
Southampton sacked Claude Puel at the end of last season after the Frenchman helped the club to eighth in the league but Pellegrino said he was not worried about his job or the prospect of a relegation battle just yet.
"When we think about the end of the last season, between eighth and 17th, there were six points. "We're talking about 10 teams and six points. The Premier League is really tough, and a lot of teams are in similar situations.
"Everything can happen or can change in one game, two games. I'm not scared about my job. I'm scared about a lot of things, not about my job.
"We have to fight for every point. We are three points from eighth, or four points away - to be in the top of the table or the bottom."
Everton are 16th in the league and come into the game on the back of a 5-1 thrashing by Atalanta in the Europa League.
(Reporting by Shrivathsa Sridhar in Bengaluru)
