Liverpool's defence looked shaky as the half wore on, with Klopp's side lucky to not be behind after Saints striker Che Adams came close on several occasions.
With the half-time whistle looming, Mane picked up the ball on the edge of the penalty area and rifled in an excellent strike into the back of the net to give Liverpool the lead against the run of play.
The Reds dominated possession in the second half and doubled their advantage 20 minutes from time as Firmino danced across the Southampton defence and sent a powerful low shot past Angus Gunn.
Despite a nervy finish to the game, Liverpool battled to earn all three points in their first away Premier League fixture of the campaign.
Slide through our gallery below for all the best snaps from Vodafone Park on Wednesday.