The home side started brightly and Jonjo Shelvey came close to breaking the deadlock when his powerful shot struck the post.
Rodgers’ men finally made the breakthrough just before half-time when Daniel Agger out-jumped the Saints’ defence and guided a header past Paulo Gazzaniga.
Southampton rarely threatened, although Rickie Lambert twice tested Pepe Reina from long range, while the Reds should have extended their lead as Raheem Sterling and Luis Suárez both came close to adding a second.
“I thought we were terrific,” Rodgers told BBC Sport.
“We just lacked more of a finishing touch. I really enjoyed watching us today.
“As the season goes on, hopefully, some of our chances will start to go in.
“Overall, we were well deserving of the win and the game should have been out of sight sooner.”
Lucas Leiva completed 80 minutes at Anfield after returning from a thigh injury picked up in a draw with Manchester City back in August.
Rodgers opted to withdraw Joe Allen in the 60th minute and introduce Jordan Henderson – and the Liverpool boss revealed he was unwilling to replace the Brazilian, who was enjoying a fine return to action.
“I thought he was excellent. He’s come so far in training, I felt I needed to get him into the games,” added Rodgers.
“He’s another piece to what we’re trying to do here and you seen the difference he makes when he’s in the team.
“He offers the team great stability and he can deal with the ball, which is important.
“My plan was to give him 60-65 minutes, but he was comfortable and it was probably only in the last 10 minutes that he started to tire, which is when I took him off.
“It was a very, very good performance for him and it’s great to see him back playing.”
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