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 was a poor day at the office. The problem is we have to live with it for a couple of weeks now."