The Reds had to come back from a goal down after Sander Berge had netted from the penalty spot in controversial circumstances.
Roberto Firmino equalised just before half-time after tapping in the rebound from Mane's shot.
And the Senegal international was involved again in the second-half when Diogo Jota grabbed the eventual winner.
On the hour-mark, Mane worked the ball down the line before delivering a pinpoint cross for Liverpool's summer signing to score his second Premier League goal for the Reds.
It win takes Liverpool up to second in the table, behind Merseyside rivals Everton on goal difference.
But all was not well, with Mane visibly upset with Salah during Saturday's encounter.
With 10 minutes to go, the Egypt international had the chance to put the game to bed but effort hit the post.
However, mere yards away Mane was ready to receive the ball and tap into an empty net.
Instead, the former Southampton star was left alone and was absolutely fuming.
It is not the first time Mane has been left incensed by Salah's selfishness in front of goal.
Almost this time last year, Liverpool had racked up a routine win over Burnley but the former Roma star was caught in a similar situation to that on Saturday.
Salah could have squared the ball to Mane to score but went for goal instead, missing in the process.
A few moments later, Mane was hauled off and he was not happy, shouting at members of Liverpool's coaching staff about the incident.
Meanwhile, Jurgen Klopp has opened up over his feelings after fielding four attackers in Salah, Mane, Firmino and Jota from the start.
He said: “We were a bit too hectic at the start. We had four offensive players on the pitch but all offered the same option.
“But I love half time. The second half started good and for 20 to 25 minutes we were really good. We could have scored more and then they came back.
“We had to dig deep but the boys did.
“It’s nice to win 2-0 or 3-0 every game but that only happens if, in games like this, we are ready to fight hard. On nights like this you earn the right to win the others like that."