Rodgers told his post-match press conference: "We just weren't very good, it's as simple as that.
Whenever you start a game the way we did, it's very, very difficult at this level to get back. We only started to play when we went 2-0 down.
"We got it back to 2-1 and were better in the second half, but then conceded another very poor goal.
"If you concede the goals we conceded today, it's always going to be difficult to win games.
"Offensively we were bright, created chances and could have scored more than one goal, but we didn't defend well enough - and you have to give credit to Southampton as well because they played well."
He said: "We've played a lot recently with three midfield players in there and it gives us the chance to control and dominate the game.
"Unfortunately this week, we had a couple of players that had niggles and hadn't had a full weeks' training - like Lucas Leiva. We put Lucas on in the second half and he was excellent.
"We just didn't quite have our rhythm today and that was across the team. We've no excuse.
"We were beaten today and we have been much better than that. Today was just a poor day at the office. The problem now is you've got to live with it for a couple of weeks and make sure at Villa we're back on form again."
On how much his side missed Carragher in the centre of defence, Rodgers said: "There's no question.
"For us, it's an area we feel we need to be better in. There's no doubt about that and that was one of the reasons for bringing him back - for his leadership qualities.
"But it wasn't the sole reason. We haven't been giving soft goals away like that for a long time now. That has given us greater confidence within the team.
"I know where we need to improve and where we need to develop as a group. Unfortunately today was a day where we didn't play as well or defend as well as we have done. I know as a coach and a manager the areas where we can improve."
Asked about Liverpool's aspirations of securing European qualification, Rodgers concluded: "There are still a number of games to go. We've still got eight games. It's just about us concentrating and keeping our focus on the next game. We can't get too far ahead [of ourselves] - there's still a lot of work to do here.
"We've had a lot of plaudits in the last few weeks because of our goals and how we've been working, but today is a reminder for us that we still need to build our squad and look to improve.
"The players have been brilliant - I can't have asked anymore from them. You can't be perfect every time you play."
