Liverpool are on 89 points after 37 Premier League matches, with leaders Manchester City on 90
Jurgen Klopp's side knew defeat would hand the title to Manchester City - while even a draw would effectively end the challenge because of their inferior goal difference.
And it looked as if Liverpool might have a struggle on their hands when Nathan Redmond gave Southampton a 13th-minute lead with a curling right-foot shot beyond Alisson.
Liverpool, with nine changes from the side that won the FA Cup by beating Chelsea on penalties at Wembley on Saturday, once again showed their resilience and reserves of strength as Takumi Minamino levelled before half-time with a thumping drive against the club where he spent a spell on loan last season.
The pressure built after the break and Liverpool took a deserved lead after 67 minutes when Joel Matip sent an instinctive looping header over Saints' keeper Alex McCarthy.
City will still retain their title with victory over Aston Villa at Etihad Stadium but Liverpool have pushed them all the way again and ensured the fight will go down to the wire.
Liverpool were under pressure going into this game - and not just because they could not afford a slip.
Manager Klopp had to make a host of changes because of tired legs following that gruelling FA Cup final win and also had to contend with injuries to major figures Mohamed Salah and Virgil van Dijk.
This was a test of the nerve as well as the depth of Liverpool's squad and, as at Aston Villa last week when they also fell behind, an examination of their ability to survive adversity and demonstrate powers of recovery.
All the tests were passed in convincing style as Liverpool dominated once they had recovered from shock of Redmond's outstanding goal.
In the absence of Salah and Sadio Mane, and with Luis Diaz on the bench, Roberto Firmino gave an outstanding display of touch and intelligent running, putting in a tireless shift that earned him a standing ovation when he was substituted late on.
Ibrahima Konate has proved to be an outstanding signing and he was once again a tower of strength in defence, ensuring Liverpool did not miss Van Dijk - his defensive partner Matip scoring the winner with what looked almost like a reflex action header, albeit a highly effective one.
The only blot on Liverpool's night was an injury to the luckless Joe Gomez, who fell awkwardly under a touchline challenge and limped off disconsolately at half-time with the consoling arm of goalkeeper Alisson around his shoulder.
Klopp delivered his tradition triple fist pump in front of Liverpool's travelling fans at the final whistle.
He will know City are still in pole position and have the title's destiny in their own hands but all Liverpool could do was put them through a final-day test and this was mission accomplished.
Southampton manager Ralph Hasenhuttl was hoping for a vibrant display from his side after feeling the full force of fan discontent following their 3-0 defeat at Brentford last time out.
The Austrian faced chants of 'you don't know what you're doing' and 'sacked in the morning' during the dismal defeat in west London on 7 May.
This was a tame response, ending as he led his team around a largely deserted stadium on a lap of appreciation after the final whistle.
Saints had the boost of that early lead but never carried the air of a team confident it would be maintained. Liverpool confirmed their obvious superiority while the home side looked nervous, especially when playing out from the back, where they played themselves into trouble on several occasions.
Southampton lie in 15th place, nearing the end of the season in which moments of promise were eventually overshadowed by disappointment. This is exactly where they finished last season and they have to win at Leicester City on Sunday to equal last term's points tally.
Now Hasenhuttl will hope to keep hold of star man James Ward-Prowse this summer and get some much-needed financial backing from the club's new owners, the company backed by Serb media mogul Dragan Solak.
-- Courtesy of BBC Sport