Joel Matip’s crucial second-half winner moved the Reds a single point behind leaders Manchester City to set up a tantalising finale on Sunday.
Jürgen Klopp’s men knew they had to be victorious on the south coast to remain in contention to become top-flight champions for only the second time in 32 years.
But they were stunned by Nathan Redmond’s superb solo opener.
However, the much-changed visitors controlled proceedings for almost the entire evening and levelled through former Saints loanee Takumi Minamino.
Then, Matip completed the turnaround with the aid of a deflection off Kyle Walker-Peters.
The Reds, who remain in contention for an unprecedented quadruple, will welcome Wolves to Anfield in five days’ time.
They will be hoping Steven Gerrard’s Aston Villa can do them a major favour away to Manchester City.
The defeat for 15th-placed Southampton was an eighth in 11 games.
Klopp bemoaned the scheduling of this must-win fixture due to it coming three days after the Reds’ gruelling FA Cup final win over Chelsea.
The German opted for nine changes from that penalty kick shootout success, with goalkeeper Alisson Becker and defender Ibrahima Konate the only men retained.
Sadio Mane joined injured duo Virgil van Dijk and Mohamed Salah in being given the night off.
His rejigged side enjoyed plenty of early possession but were rocked by Redmond’s fine 13th-minute effort.
After Lyanco dispossessed Diogo Jota just outside Saints’ box, Redmond received the ball from Nathan Tella and ran almost half the length of the pitch.
He then cut in from the left and delightfully curled home from the edge of the area via a slight deflection off James Milner.
Klopp felt the robust challenge on Jota was a foul and he had cause for further frustration minutes later.
This was when Roberto Firmino’s precise headed finish from a Kostas Tsimikas free-kick was flagged for offside.
Choruses of “God Save the Queen” rang out from sections of the home support.
This was in view of the fallout of Liverpool fans booing the national anthem at Wembley at the weekend before the visitors soon levelled.
Japan international Minamino claimed the 27th-minute equaliser.
He beat recalled Southampton goalkeeper Alex McCarthy at his near post with a powerful, rising drive into the roof of the net after being expertly slipped in by Jota.
Liverpool were forced into a change at the break as captain Jordan Henderson replaced the injured Joe Gomez, pushing the versatile Milner to right-back.
The enforced alteration had little impact on the flow of the game and, after going close through Jota and Harvey Elliott, the visitors decisively edged ahead in the 67th minute.
Tsimikas’ outswinging corner-kick from the left was inadvertently flicked on by Saints winger Mohamed Elyounoussi.
Matip held off Walker-Peters to send a looping header beyond McCarthy and into the right corner in front of the delirious travelling fans.
Southampton rarely threatened a costly equaliser, although Liverpool goalkeeper Alisson fumbled a Redmond effort from distance late on.