The Reds arrived at St Mary's on the back of four successive wins but were second best from start to finish.
"We just weren't very good. It was as simple as that," Rodgers said. "If you start a game the way we started it then it is very difficult at this level to get back."
He went on: "We only started to play when we were 2-0 down. We got it back to 2-1 and we were better in the second half but then conceded another very poor goal.
"(It) was a poor day at the office. The problem is we have to live with it for a couple of weeks now."
Rodgers was most aggrieved at the way his side defended.
"When you concede the goals we conceded it is always going to be difficult to win games," Rodgers said. "It's something I know that we need to improve. We need to develop as a group. We didn't defend or play as well as we can. I know the areas where we can improve."
Southampton's win was just their second in the league under manager Mauricio Pochettino, and he said: "I think we set out the game in a very positive way.
"Most importantly the players believed in the way we set up. I think we made Liverpool really uncomfortable on the pitch.
"We pressed them quite a lot and we put in a really big effort. I think my team played really well."
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