The victory, secured by goals from Coutinho and Raheem Sterling, means Liverpool have not tasted league defeat since going down to United in December and leaves the dark old days of four consecutive losses last November very much in the past.
The Reds are now sixth, only two points adrift of United who occupy the last Champions League spot, and with matches against City at home (next Sunday), United at home and Arsenal away all to come in the month ahead.
“The first part of the season was nothing like we’ve been, so we had to find a solution,” said Rodgers. “We’ve restructured the team in a different way to get that solidity back and the players’ response has been magnificent. Today was a huge win; to come here on the top of a tough European game and to win and not concede was excellent.
“We are ready for City and the players will relish every game now. It’s a big step to be where we are considering how far we were behind. Now we’ll use the experience of last year’s run-in and keep the mentality, keep the focus and keep the performance up until the end.”
It is hard to argue against Liverpool’s impressive recovery, and with Emre Can at left centre-back and Adam Lallana and Dejan Lovren being roundly booed on their return to St Mary’s, they did well to complete what in the end was a comfortable victory, even if referee Kevin Friend also made a telling contribution in the opening 45 minutes.
Friend, who was also the fourth official on Saturday when Jose Mourinho fumed at decisions that went against his side, turned down three Southampton penalty claims, the first two extremely convincing.
Friend then failed to send off Simon Mignolet (or even award a free-kick) when he raced out of his area to deny Saints winger Elia and appeared to handle the ball in the process.
Liverpool were pretty poor in a first half played in driving rain and a swirling bitterly-cold wind, but still led at the break thanks to a quite wonderful goal from Coutinho, which was quite out of context with the quality of the game.
He picked the ball up in space almost 30 yards from goal and sent a beautifully swerving effort right into the top corner with the precision of an artist.
The second half, in fairness to Friend, was less controversial, but also lacking in creative spark, as Southampton pushed for an equaliser without ever coming that close.
The visitors, marshalled well in midfield by Joe Allen, were not at their best but nevertheless secured victory with a second goal courtesy of two poor mistakes, not by the referee but by Southampton players.
First the ball was given away by Schneiderlin, allowing Liverpool to break, and then young left-back Targett mishit his clearance straight to Sterling, who made no mistake from 10 yards.
Southampton manager Ronald Koeman steered clear of criticising referee Friend, although he did suggest the second penalty claim was a clear one, and instead explained his decision to leave in-form striker Sadio Mane on the bench.
“He was left out because he was 25-30 minutes late. We have rules. Everybody can be late one morning but you can’t be late for a match against Liverpool. He had reasons and that’s between the club and the player, but I don’t accept that. The other players did a good job without him. We could have fined him but in my opinion the sanction for him not to play is the best way to keep the discipline in the team.
“As for the game, we were unlucky about the decisions of the referee in the first half and they scored a great goal; and then it’s always difficult. But I was proud of the way my players played.”
Rodgers can certainly echo that sentiment about his own squad; and with Liverpool right on Southampton’s shoulders, they go into a run of big matches against big opponents in the best possible shape.
SOUTHAMPTON: Forster 6; Clyne 7, Fonte 7, Yoshida 6, Targett 6; Wanyama 6, Davis 6 (Schneirderlin 46), Ward-Prowse 5 (Mane 57, 6); Djuricic 7 (Tadic 75, 6), Pelle 6, Elia 6.
LIVERPOOL: Mignolet 6; Can 5, Skrtel 7, Lovren 6; Ibe 7 (Johnson 75, 6), Henderson 6, Allen 7, Markovic 6 (Moreno 46, 6); Lallana 5 (Sturridge 63, 6), Coutinho 7, Sterling 7.
Referee: Kevin Friend.