Salah replaced the injured Joe Gomez, getting a big applause from the Anfield crowd for his early introduction. Dominik Szoboszlai, who had started on the right wing, moved to right-back to facilitate Salah's entry.
The Egypt international made his impact known in the second half with a well-delivered corner that Hugo Ekitike nodded home to make the score 2-0.
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It meant Salah moved ahead of Wayne Rooney to become the Premier League's all-time leader in combined goals and assists for a single club with 277.
Salah had a chance to add a goal of his own in stoppage-time when Federico Chiesa slid the ball to him across the box, but his first-time shot soared over the bar.
Salah had been recalled to the matchday squad after being left out of Tuesday's Champions League win over Inter Milan.
The move came after Salah gave a rare interview to news journalists after the draw to Leeds United when he was dropped to the bench for a third straight game. His most explosive comment was saying he felt like he had been "thrown under a bus" by the club.
This was Salah's final game for Liverpool before heading to the Africa Cup of Nations.
Salah had suggested last weekend that the Brighton clash could be his last for Liverpool, although Slot said this week that he does not want Salah to leave in January.
Premier League combined goals/assists for single club
1. Mohamed Salah, Liverpool - 277
2. Wayne Rooney, Manchester United - 276
3. Ryan Giggs, Manchester United - 271
4. Harry Kane, Tottenham Hotspur - 259
5. Thierry Henry, Arsenal - 249
6. Frank Lampard, Chelsea - 237
7. Sergio Aguero, Manchester City - 231
8. Steven Gerrard, Liverpool - 212
9. Son Heung-min, Tottenham Hotspur - 198
10. Jamie Vardy, Leicester City - 193
