Jurgen Klopp's men had the majority of possession, but Southampton goalkeeper Fraser Forster denied his former team-mate Sadio Mane in the first half and Philippe Coutinho, Roberto Firmino and Nathaniel Clyne missed second-half chances.
Liverpool remain in first place ahead of Manchester City only on goals scored, but now Chelsea will go above them with a win at Middlesbrough on Sunday.
Both sides were without key creative players, with Southampton midfielder Dusan Tadic not risked after breaking his nose while on international duty.
Former Southampton captain Adam Lallana did not travel with the Liverpool party after sustaining an injury while playing for England.
But Coutinho was deemed fit enough to start despite a hamstring problem while Dejan Lovren, another ex-Saint, returned to the back four after recovering from a virus.
Lovren was jeered every time he received the ball in an early spell of Liverpool possession that forced the home side back towards their own penalty area.
But when the Merseysiders were able to find a gap, either their touch or the final pass eluded them.
Southampton were seeing little of the ball and were restricted to attacks on the break, one of which ended with right back Cedric Soares shooting a yard wide with his left foot from 22 yards.
The Portugal defender then sent over a cross from the right that Liverpool goalkeeper Loris Karius punched clear under very little pressure.
Appealed in vain
Otherwise it was mostly Liverpool and they created what could have been a decisive opening in the 28th minute when Georginio Wijnaldum found Mane free in the penalty area with an intelligent diagonal pass.
Mane's shot was struck hard and true but Saints' England goalkeeper Forster flung himself to his left to save.
Five minutes from the break, Mane had another chance as he charged goalwards from the right but Virgil Van Dijk slid in to block the Senegal winger's shot.
Saints were still not an attacking force, which was shown when Charlie Austin got into the penalty area and looked around for support -- but found none.
The second half opened in similar fashion. Liverpool appealed confidently but in vain for a penalty as Van Dijk tugged Firmino's shirt, Forster saved Mane's shot at the foot of his near post and Coutinho sliced his shot wide after Firmino's pass sent him in on goal.
Then it was central defender Joel Matip's turn to miss chances. First he failed to reach Coutinho's inviting free kick from the right with an attempt at a header and the result was the same from a corner from the other side.
Liverpool's wastefulness almost cost them after 59 minutes when Austin's head met a cross from Soares but his effort glanced just wide.
Five minutes later Liverpool missed their best chance yet. Coutinho's pass found Firmino advancing into the penalty area with only Forster in front of him, but he slid his shot wide from 10 yards.
Next, Clyne headed wide after beating teammates and opponents alike to a cross from substitute Daniel Sturridge.
That was to be the final chance as Southampton settled for a draw after successive defeats by Chelsea and Hull.