Mauricio Pellegrino recognises Southampton had little choice but to sack him in their bid to preserve Premier League status.
The Argentinian, 46, was dismissed on Monday evening with the club 17th, only a point above the bottom three, and with just eight remaining league fixtures.
It's understood that they hope to appoint a successor by Sunday's FA Cup quarter-final at Wigan and that their intention is to find a long-term replacement, not an interim one as other clubs in their position often consider.
Mark Hughes, until his sacking in January the manager at struggling Stoke, is the present bookies' favourite to be appointed.
But 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 finish ed. 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 everyt hing required to move forward."
Southampton's next Premier League fixture is at relegation-threatened West Ham on March 31. Should they not succeed in appointing a new manager in time for the trip to Wigan, their priority will be on ensuring he has instead arrived by then.