Daniel Agger's 43rd-minute header came at the end of a first half the home side dominated entirely.
The closest they came to opening the scoring before the goal was when Jonjo Shelvey smashed a drive against the post but a host of other chances went begging.
"The game should have been put to bed well before it was," said Rodgers.
"I thought we were outstanding the first half.
"The tempo, we shared the ball, everyone was passing and moving, the creativity [was good] in the team throughout the game and it was just getting that little bit of luck with the final ball.
"No matter how much you dominate the game, if it's still only 1-0 going into the final 10 minutes it can always offer you a bit of challenge where you have to defend.
"There's no question we should have been out of sight, but the most important thing is it was a terrific performance and another win at Anfield.
"We continue with another three points and moving up the league."
The one downside was a booking for leading scorer Luis Suarez when he needlessly tried to punch in Steven Gerrard's cross, meaning he will be suspended for next Sunday's trip to West Ham.
"That's the way it goes. That was always going to be the risk, but it's an opportunity for someone else to come in," said the Reds boss.
Southampton manager Nigel Adkins felt they could have got something from the game, with Rickie Lambert having threatened with two first-half shots and an improved performance after the break.
"We are disappointed in respect of we haven't got anything from the game," he said.
"We have come away from home in a good run of form of late but Liverpool's movement was very good today [Saturday] and we couldn't get a grip of the football.
"If we could have got to half-time at 0-0 that would have been pleasing but a free header in the penalty area has given them a 1-0 lead.
"We had to adapt ourselves a little bit. We have defended a lot deeper and looked to counter-attack.
"Credit to the players because we managed to hang in there and the last 10-15 minutes we went a lot more direct."