Raheem Sterling opened the scoring midway through the first half as he latched onto a defence-splitting through ball from Jordan Henderson, but Nathaniel Clyne surged into the box to equalise as Southampton threatened an upset.
However, Daniel Sturridge clinched the three points 11 minutes from time when his deft touch from close range guided Sterling's return header beyond debutant goalkeeper Fraser Forster.
The hosts did survive a late scare as Simon Mignolet tipped Morgan Schneiderlin's pile-driver onto the bar before Shane Long headed the rebound wide from six yards, but they managed to hold off the late pressure to secure the win.
Rodgers told Sky Sports: "It was very similar to last season - the first game of last season (Mignolet saved a stoppage-time penalty as Liverpool edged past Stoke 1-0).
"Southampton played very well in the first half and we had to dig in, we were obviously not anywhere near our usual fluency in the game.
"We scored two very good goals and we're delighted about the three points.
"I think in terms of the character and desire and the will to win we were exceptional. We made blocks when we had to.
"Obviously towards the end when you're 2-1 up and they're throwing everything at you, you've got to be strong and brave and make challenges.
"I thought Simon Mignolet made an incredible save to touch it on to the bar. He saved a penalty last year so it was very similar to that game but we showed the improvement in our mentality today to get the three points.
"That was important today and as the season goes on, we'll get better and better."
Summer signing Rickie Lambert was introduced for his Liverpool debut against his former side and Rodgers hailed the strength in depth now at his disposal after a busy transfer window.
"I thought Rickie Lambert came on and made a difference to us. He offered that threat, that extra body down the middle when we changed to a diamond and Dan got a great touch to score that second goal," he said.
"We've got a few more to come back, we've got the likes of Adam Lallana who's a great talent and young Markovic who's an exceptional young player as well.
"I felt we could do something today to change the game and he was able to come on and cause a problem. When he came on the pitch we started to cross it a bit more."
Asked if it would take time for his side to gel, Rodgers replied: "Not really, no. We had Luis Suarez here last year and it was a similar sort of game. The most important thing was to win and we showed the capacity to dig deep today.
"Southampton played well, you've got to give them credit - they've got good players but the important thing was for us to get the three points."
Rodgers also praised Man of the Match Sterling and goal-hero Sturridge for picking up from where they left off last season to play influential roles.
"Those boys did that last year so I don't think they're stepping up, I think they're continuing at the level they were at last season," he said.
"I think it's 36 goals for Daniel in 50 appearances - I think he's the most clinical striker now for the club in terms of the first 50 games.
"For him to get that goal was very important. Young Raheem's was great, Jordan Henderson won the ball back in that transition moment and played a great pass for Raheem.
"There won't be many. We know where we want to add. At this moment, the group are up and running and we're looking forward to the next game."