On Monday night, Southampton announced that they had parted company with Pellegrino after a run of just one win in 17 matches in the Premier League had left the club just one point above the relegation zone.
Pellegrino leaves St Mary's having won eight of his 34 matches in charge, but the Argentine has acknowledged that he cannot dispute being sacked by the south coast club.
In an open letter published by the Daily Echo, the 46-year-old said: "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.
"I am really happy with the hospitality of the people of Southampton and the respect you have shown me in every moment during this experience. The fans have been behind us even in difficult games and I am really grateful for your support in those situations.
"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. 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."