Jurgen Klopp's side were in cruise control at St Mary's after Firmino broke the deadlock.
Their victory, coupled with United's shock defeat at Newcastle, cut the lead enjoyed by Mourinho's outfit over Liverpool to just two points, while defeat left Southampton rooted in the relegation mire, a point from safety.
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The two-goal lead was harsh on Southampton going into half time, but it nearly got worse in the opening stages of the second half when Mane's acrobatic effort was blocked and fell at the feet of Salah, although this time the prolific winger scooped his finish over the crossbar. It was cold and blustery on Sunday and Southampton's defenders were as physical as referee Martin Atkinson would allow, but Firmino appears to relish such circumstances. The whole Southampton team have only scored seven more league goals than Salah has in 2017-18. But the turn of the year saw his side pick up results that keep have taken some of the heat off, for the time being at least. Liverpool have only lost one match (away to Swansea) in their last 17 (11 wins, five draws) Premier League games.
Mane and substitute Adam Lallana had further opportunities in the final 15 minutes, though neither miss mattered a great deal as Liverpool clinched all three points with minimal fuss.
The German goalkeeper was then put to work by headers from Guido Carrillo and twice by James Ward-Prowse but wasn't unduly troubled.