Southampton dismissed Pellegrino last night after just one victory in 17 league games which leaves the club dangerously close to the relegation.
Saturday’s 3-0 defeat at Newcastle leaves the Saints just a place and a point above the bottom three and sealed the Argentine’s fate.
Le Tissier reckons the turmoil surrounding Van Dijk’s future and the persistent links with Liverpool throughout last summer affected the team’s performances early in the season.
Van Dijk eventually moved to Anfield in January for £75m to finally conclude the lengthy, and at times ugly, transfer saga.
Le Tissier, who scored 209 goals for Southampton during his 16 years at the club, said: “I think that was probably something that, looking back with hindsight, the club would probably regret, trying to keep him in the summer.
“They tried to do the right thing by the fans, in terms of not selling their best player every season, but sadly that did appear to backfire, and that situation probably didn’t help the performances in the first half of the season.”
Southampton have lost a large number of players to their domestic rivals in the past few year.
Saints chairman Ralph Krueger said earlier in the campaign that they were keen to end the perception that they were a selling club.
Le Tissier added: “That’s life, that’s football, those decisions were made with the best intentions and it didn’t turn out right.
“It’s now time to move on, appoint a new manager, and get out of the trouble we are in at the moment.”
Southampton will now begin their search for a new boss with Marco Silva and Brendan Rodgers among the early favourites.
The club said in a statement: “We would like to place on record our thanks to Mauricio, [Assistant Manager] Carlos [Compagnucci] and [Assistant First Team Coach] Xavier [Tamarit] for their efforts during their time with Southampton, and wish them well for the future.
“The club will look to appoint a new management team as soon as possible, with the search for a replacement already underway.”