Salah scored for the first time in the Premier League since Boxing Day as Jurgen Klopp's men snapped a four-game winless streak.
Cody Gakpo then grabbed his first goal for the club in the second half.
A positive night for Liverpool was rounded off by the return of Diogo Jota and Roberto Firmino from injury, while Virgil van Dijk made the bench after six weeks on the sidelines.
"It's a huge win for us," said Salah. "We had a perfect week to train and the players were so excited and we couldn't wait for the game to turn everything around. Hopefully it was a start."
Gakpo will hope a derby goal proves the turning point after a difficult start to life in the Premier League following his January move from PSV Eindhoven.
"It was a great game from our side," he said. "We really needed this."
Liverpool are still nine points off the top four in ninth.
But that gap to fourth-placed Newcastle could be cut to six points when they travel to St. James' Park next weekend.
That clash comes just days before Real Madrid's trip to Anfield in the Champions League last 16 and Klopp will be hoping he has finally stumbled upon a solution to some of his side's woes in time for another tilt at the European Cup.
"Tonight we were there and that has to be the sign now for us for what we have to do," said Klopp. "The performance for the full 96 minutes was the best for a while."