MacAnthony has since appointed former Motherwell boss Jim Gannon, after Cooper managed just one win in 13 games during his 10-week spell at the club.
He told the club website: "Mistakes get made, but it's not how you make them that matters, it's how you fix them.
"I take full responsibility for the recruitment and departure of Mark."
He added: "I will always hold my hands up when something does not work out because of a decision that I make.
"After losing eight times in 13 games, including five defeats in January where we conceded 12 goals and only score one, his position became untenable and I felt it was time for a change for the good of the club."
Posh, who currently lie bottom of the Championship 11 points from safety, face Queen's Park Rangers on Saturday.
"There haven't been a lot of great times for us in the Championship and our confidence is at a serious low point," said MacAnthony.
"But I have appointed the right manager to take us forward and install some much-needed confidence all over the club, top to bottom, and have a good go at these last 18 games.
"Hopefully we can get back to winning football matches and playing attacking football once more and unite our dressing rooms and of course our fans."
Source: BBC Sport
>> Terms & Conditions