Jurgen Klopp's Reds started poorly on the south coast, seeing their hosts squander a couple of presentable first-half chances, but Mane, as so often, came up with the goods.
His brilliant effort — a third of the season after two in the UEFA Super Cup final — woke Liverpool from its slumber and it was comfortable in the second period even before Firmino doubled the advantage.
There was a wobble when Adrian hacked the ball straight at Danny Ings for an unlikely Saints response. Former Reds striker Ings should have equalised with a gilt-edged chance for his second, but Liverpool held on.
A run of nine wins to close out last season could not deny Manchester City the title, but that streak of league successes goes on — reaching a club-record-equalling 11 — as Klopp's men look to be in the mix again this term.
Liverpool, perhaps missing Alisson's calm in possession, looked shaky at the back early on, but was grateful to Adrian — passed fit after a freak ankle injury in the Super Cup celebrations — for an instinctive close-range stop from Maya Yoshida after a corner.
The hosts remained on top and had another huge headed chance as Che Adams met Oriol Romeu's cross 10 yards out but nodded over the crossbar.
But after 46 minutes of toil, former Southampton man Mane came up with a moment of inspiration for the European champion in first-half stoppage-time, cutting inside from the left and drilling into the top-right corner from 20 yards.
Liverpool was improved after the restart and Joel Matip flicked James Milner's scuffed shot agonisingly wide, before Angus Gunn denied Mohamed Salah as he raced through on the right.
Firmino should have stretched the lead when he slid a finish past the right-hand post, but he made no mistake the next time he was granted a sight of goal, moving from left to right in the area and striking low into the bottom-left corner.
Gunn expertly saved from Mane and Andy Robertson, and Southampton threatened a late fightback when Adrian comically hurried a clearance into Ings' shins with the ball finding the back of the net. The former Liverpool man squandered another late chance from close range as Klopp's men escaped.
Liverpool is now already four games into the new season and its next task is a tough one, with Klopp's men hosting Arsenal next weekend. Southampton goes to Brighton and Hove Albion.