Pellegrino was appointed at St Mary's at the start of the season, taking over from Claude Puel after impressing with Alaves in LaLiga.
In a statement posted on Southampton's official website, Pellegrino addressed the club's fans, saying: "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 have to say thanks to [chairman] Ralph [Krueger], to the board, to [vice-chairman] Les [Reed] and [director of football] Ross Wilson, all my technical staff and all the Southampton employees. Also to my players, who respected me in every single moment.
"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."
Southampton, yet to appoint Pellegrino's replacement, travels to Wigan Athletic in the last eight of the FA Cup on Monday (AEDT), before a key away league clash with West Ham on 32 March (AEDT).