The Reds were forced to withstand consistent pressure from the hosts during the first half, but Mane's strike on the stroke of half time turned the game in Liverpool's favour.
The visitors survived another late scare to escape with all three points, though, making it two wins from two at the start of the new season and a club record-equalling 11th successive victory in the Premier League.
Liverpool's pre-match injury concerns were eased by Adrian's recovery from a freak fan-inflicted injury during their Super Cup celebrations, and initially the Reds did not look as though they were feeling the effects of their penalty shootout exploits in Istanbul.
However, Southampton soon settled and began to cause Liverpool problems, the first of which was of the Reds' own making as Adrian took too much time over a clearance and was fortunate to see the ball bounce behind after James Ward-Prowse had blocked it.
Che Adams then got in behind the Liverpool defence to threaten again, with Adrian needed to thwart the summer Southampton signing.
Alex Oxlade-Chamberlain - starting his first Premier League match for more than a year - dragged an effort well wide at the other end following a clever flick from Mohamed Salah, but it was a rare moment of attacking promise for Liverpool during a half in which they were largely second best.
Adams flashed a volley narrowly wide which Jannik Vestergaard and Maya Yoshida were both close to turning on target before Yoshida did get an effort on goal from a corner, only to be denied by a stunning save from point-blank range by Liverpool's Super Cup hero Adrian.
Liverpool did begin to stem the flow of chances and almost caught Southampton on the counter-attack shortly past the half-hour mark when Salah latched on to a pass over the top before flashing a shot off target.
The best chances continued to fall the way of Southampton, though, and the hosts should have taken the lead seven minutes before half time when Adams nodded just over from Oriol Romeu's cross.
Southampton's profligacy would come back to hurt them on the stroke of half time when Liverpool showed their clinical side to take the lead against the run of play, with Mane adding to his brace in midweek.
The former Saints forward collected a pass from Milner just inside the area before opening his body up and curling a fine effort into the far corner which left Gunn helpless.
The goal had a galvanising impact on Liverpool at the start of the second half as they enjoyed a far greater degree of control over the game, and Milner almost added a goal to his assist 10 minutes after the restart when his low drive flashed narrowly wide.
Salah had an even better chance moments later when Mane sent him clean through on goal with only the goalkeeper to beat, but he put his finish too close to Gunn when trying to pick out the bottom corner.
Milner then fired an effort over the crossbar before another golden chance went begging for Liverpool when Mane slid the ball into the path of Firmino, who opened his body but passed his finish wide when he should have scored.
Southampton could not punish Liverpool's wastefulness in the same way as their own had been, though, with Ings coming close when he cut inside before curling a long-range effort narrowly over.
Liverpool's second 19 minutes from time appeared to have put the game to bed as Firmino made up for his earlier miss, collecting the ball from Mane and biding his time before firing his finish past the wrong-footed Gunn.
The goal saw Jurgen Klopp's side equal their club record of scoring multiple goals in 11 consecutive Premier League games, but they were in no mood to settle for just two and it took some smart saves from Gunn to avoid further damage for the hosts.
First Mane was denied with a header at the near post and then Andrew Robertson forced another stop moments later following slick link-up play with the irrepressible Mane.
Just when it looked as though the Reds would cruise to a routine win, Southampton were handed late hope courtesy of a howler from Adrian, who kicked the ball straight at Ings when trying to play out from the back despite having ample space to pick out either side of the Southampton striker.
It was a bizarre way for Southampton to score their first goal of the season - and Ings his first of 2019 - but the former Liverpool striker should have added a second to rescue a point for the Saints just three minutes later.
Yan Valery played a perfectly-weighted low pass into the box which Ings simply had to score, only to miss his kick as the ball bounced across the six-yard box and out for a goal kick.
It proved to be the final opening as Southampton were left to rue their missed chances, leaving them still searching for their first point of the campaign whereas Liverpool have quickly picked up where they left off last season as far as results are concerned.
SOUTHAMPTON (3-5-2): Gunn; Bednarek, Yoshida, Vestergaard; Valery, Ward-Prowse, Romeu (Ings 64'), Hojbjerg, Bertrand (Djenepo 77'); Adams (Armstrong 68'), Redmond
LIVERPOOL (4-3-3): Adrian; Alexander-Arnold, Matip, Van Dijk, Robertson; Oxlade-Chamberlain (Henderson 89'), Wijnaldum, Milner (Fabinho 74'); Salah (Origi 79'), Firmino, Mane