Southampton manager Mauricio Pellegrino claimed the "biggest defeat" his side suffered in their loss to Liverpool was "the character we showed in the second half".
Liverpool moved to within two points of second-placed Manchester United after goals from Roberto Firmino and Mohamed Salah secured a comfortable victory.
Brazil forward Firmino gave Liverpool an early lead with his 20th goal of the season when he finished off a counter attack after being teed up by Salah.
Southampton then twice brought saves out of Reds goalkeeper Loris Karius, before Liverpool added a second through Salah after he exchanged passes with Firmino.
It was the Egypt international's 29th goal of the season for the club from 36 appearances.
Liverpool looked the more likely to score again after half-time as Southampton rarely threatened.
"The second half was really bad. The people see our reaction on the pitch and the players feel the reaction," Pellegrino said.
"The anxiety in my players was really high. We understand the emotions of the fans but we have to keep going."
The win ensured Liverpool took advantage of Manchester United's 1-0 defeat by Newcastle earlier on Sunday and leapfrogged Tottenham back into third place.
Southampton drop into the relegation zone, a point behind Huddersfield who beat Bournemouth 4-1, with further scrutiny of Pellegrino's position likely.
Liverpool defender Virgil van Dijk was expected to get a hostile reaction from the Southampton fans on his first return to St Mary's since his move to Anfield, and there were plenty of boos for the Dutchman.
Indeed, Reds manager Jurgen Klopp revealed in the build-up to the match he would speak to Van Dijk about the need to maintain his composure in light of the ill-feeling towards him.
The 26-year-old, whose protracted move to the Reds for £75m from Southampton was eventually completed in January after the transfer saga of last summer, more than kept his cool at the heart of Liverpool's defence.
His performance was one with which the Southampton fans would have been familiar - towering headers, key interceptions and brave blocks while marshalling the backline.
It was this kind of faultless defensive display Liverpool fans had yearned for before his arrival, and the kind Southampton now so desperately miss.