The Argentinian, 46, was sacked on Monday evening with the club 17th, only a point above the bottom three, and with just eight remaining league fixtures.
In an open letter on Southampton's official website, Pellegrino said: "I feel so sorry to leave the club in this situation.
"But when you don't get results I understand the decision the club has to take.
"As you know, my period in Southampton has finished. From the beginning I tried to do my best every single day, I tried to give back to the club and tried to represent the club with pride.
"Some moments we managed this, some moments we couldn't, but we have been living a difficult season for many reasons.
"The manager is the face of a lot of decisions and actions, but I always tried to put the club first with my technical staff and the board behind me.
"The fans have been behind us even in difficult games and I am really grateful for your support in those situations.
"I have to say thanks to Ralph (Krueger, club chairman), to the board, to Les (Reed, executive director) and Ross (Wilson, director of recruitment and scouting), all my technical staff and all the Southampton employees.
"Also to my players, who respected me in every single moment.
"I wish everyone all the best for the rest of the season, I have enjoyed an amazing experience and I know that Southampton has everything required to move forward."