The manager was sacked following the draw with Everton on Sunday, and Carragher has criticised the club's owners.
The manager was relieved of his position on Sunday evening following his side’s 1-1 draw with rivals Everton in the Merseyside derby, a result which left the Reds tenth in the table.
It has been a disappointing start to the campaign once again for the side with them having won just three games all season, with disappointing defeats to West Ham and Manchester United particular low points.
That followed a similarly disappointing end to last season, and Carragher ha suggested he was lucky to keep his job this summer.
“It hasn’t gone well for him and he needed to make a good start this season,” he told Sky Sports.
“He was maybe fortunate to stay on at the end of last season. I don’t like doing it now. Do it in the summer if you want to make a change. He’ll have known himself that he had to make a good start and he hasn’t.”
Owners must take portion of blame at Liverpool
The Anfield legend though, says that the manager cannot argue about his dismissal because his record has not been good enough.
During his three years in charge of the club, Rodgers failed to win a trophy with the side, while they have also only managed to reach the Champions League on one occasion with them ultimately exiting at the group stage of the competition.
Carragher though, has criticised the club’s owner also, saying their track record in charge of Liverpool has not been good enough.
“At this moment the owners' track record over the last two or three years is not good enough. It’s miles off,” he added.
“Rodgers can’t argue in some ways. He’s been there three years, he hasn’t won a trophy and they’ve played Champions League football once. That’s not good enough for Liverpool.”