Saints face third-placed Chelsea at Stamford Bridge, hoping to bounce back from their embarrassing 4-1 defeat at home to Leicester on Wednesday.
But Pellegrino has claimed Southampton will maintain a businesslike approach with the ex-Saints manager, who guided the south coast club to an eighth-place finish in the Premier League and the league cup final last term.
Yet he was deemed surplus to requirements.
Since he replaced Craig Shakespeare at Leicester, the Foxes has been reinvigorated, and Puel's free-scoring side convincingly put Southampton to the sword.
Manchester City's relentless winning streak means Chelsea appear to be in a fight for the top four, with Tottenham and Liverpool beneath them and four points behind.
Chelsea have been a little inconsistent this season and that is the main reason why they are nearly out of the picture in relation of the Premier League title race.
Puel, however, was keeping it low-key, perhaps one of the reasons he never truly won over the Southampton fans.
Okazaki wasn't the only player to come in and score, as King also got on the scoresheet at the back post following Harry Maguire's cross.
"He scored two goals and that is fantastic for him, but I am happy with all the players".
With Andy King putting Leicester 3-0 ahead before halftime boos rang around the stadium from the home fans.
But the same can not be said of Leicester, who have begun to display glimpses of their sensational 2015-16 title-winning form since Puel replaced the sacked Craig Shakespeare in October. "After conceding the first goal they were getting better and little by little our confidence went down".
"I saw him last season in the under-21 games and he can play in different positions for me", he told the Leicester Mercury.