The Reds arrived at St Mary's on the back of four successive wins but were second best from the moment Morgan Schneiderlin opened the scoring on six minutes.
A deflected Rickie Lambert free-kick inflicted further damage before Philippe Coutinho's injury-time strike offered signs of life in the visitors.
But there was no Reds fightback as Jay Rodriguez was allowed to waltz into the penalty area unopposed to seal the points 10 minutes from time.
"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.
"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.
"Today [Saturday] 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.
Southampton repeatedly harassed the Reds midfield, forcing Rodgers to hook Joe Allen at half-time, and it served to unnerve them into some haphazard defending.
They were also cut apart down both flanks on numerous occasions during the first half, when Southampton could have scored more than the two they managed.
"When you concede the goals we conceded today [Saturday] 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. Today [Saturday] 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.
The other win was against champions Manchester City last month, with Saturday's success similar in the manner it was achieved.
On both occasions Saints profited from pressing their more-renowned visitors, and Pochettino praised his players as they "neutralised" the likes of Steven Gerrard and Luis Suarez.
"I think we set out the game in a very positive way," he said.
"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.
"That's the way that we want to play. We wanted to play really high up the field in their half.
"We knew that their attacking prowess is based on the threat of Gerrard, Suarez and Coutinho and we neutralised that threat.
"I think today [Saturday] my team played really well."
The win moves Southampton seven points clear of the relegation places but Pochettino refused to be drawn on whether his side might now be safe.
"I don't really know," he said.
"What I am really focused on is preparing well for the next game against Chelsea and getting the three points.
"We played well today [Saturday] but I think the result made it seem like we have played better than we have in previous games.
"We have lost games when the team has played as well as they have done today [Saturday].
"In football it is really important to be effective.
"Today [Saturday] we had a lot of goalscoring chances. We scored three goals, two more than Liverpool, and that led us to victory.
"I always like to get victories playing under our philosophy."