Brighton had beaten Manchester United in round three, but the Seagulls have been in miserable form since and were easily beaten by an improving Liverpool at Anfield.
Curtis Jones was forced to start in an unusual role at right-back due to a number of defensive injuries for the English champions and burst into the box to open the scoring from Milos Kerkez's cross.
Salah has been a shadow of his former self for most of the season, but a second-half goal and assist could prove a major confidence boost.
The Egyptian's cushioned pass set Dominik Szoboszlai clear to hammer in Liverpool's second before Salah converted his 252nd goal for the Reds from the penalty spot.
Around the FA Cup's fourth round
Earlier, Sandro Tonali scored twice as Newcastle came from behind to beat 10-man Aston Villa 3-1.
Elsewhere, third-tier Mansfield produced the upset of the day by winning 2-1 at Premier League Burnley to reach the fifth round for the first time since 1975.
Manchester City eased past Salford 2-0.