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.
Liverpool have been particularly strong on their travels this season and have scored a league-high 34 goals.
Firmino returned the favour from the first goal with an audacious backheel into Salah's path and he slotted the ball into the bottom corner for his 22nd league goal of the campaign to close back to within one of Harry Kane in the golden boot race.
The Saints had gone 12 league games without a win before last weekend.
"I didn't know United had lost before the game, but we know we're not far away, and we play against them at least one more time this season", said Klopp.
The Saints haven't beaten a top-six side in the league now for over 18 months and the Reds ran ragged last time out and have a lot of quality in their ranks. I'm not bothered. What's important is what the manager says because he is the man in charge of me.
The pace that Liverpool have could cause a real issue for out back-line with Salah reeking havoc in the reverse fixture!
Both sides need the points but I'd like to think that as we're at home and are staring relegation in the face still that the players really go for a win. We had a little chat before the game, asking him what he thought.