Arne Slot's team quickly turned the game around with two goals in just three minutes during the second half, overturning Brighton's early lead given by Ferdi Kadioglu.
Seeking revenge for their midweek Carabao Cup defeat to the same opposition, Brighton dominated the early stages. Kadioglu's breakthrough goal was one that involved 21 passes before he drilled his side into the lead with just 13 minutes played.
The visitors continued to trouble Liverpool, who struggled to gain any kind of control as Brighton troubled their back four. Georginio Rutter came close to doubling the Seagulls' lead before the hour mark but drew out a vital one-on-one save from Caoimhin Kelleher.
Goalscorer Kadioglu nearly netted his second as Brighton broke with Liverpool's defence back peddling but he watched his volley at the back post flash over the bar after meeting a chipped Kaoru Mitoma cross.
TrendingAs the first half drew to a close, a rare Liverpool first-half attack saw Ibrahima Konate fall heavily at a corner and forced off during the interval with his arm tucked into a makeshift sling.
After the break, Liverpool swung the momentum, finding rhythm in their passing as they searched for an equaliser. All of a sudden, Bart Verbruggen found himself very much involved in the Brighton net, keeping out Alexis Mac Allister with a reaction save minutes after the restart.
Following continued Liverpool pressure, Gakpo followed up his brace in midweek with the leveller before Salah whipped in the winning goal.
Victory for Liverpool means they find themselves top of the league two points ahead of Manchester City who suffered defeat at Bournemouth. Defeat for Brighton denied them a place in the top four and leaves them in seventh.
Liverpool boss Arne Slot:
"We were outplayed in the first half, they were better on the ball, more aggressive without the ball and everything changed in the second half. The second half was completely the opposite. The main difference in the second half is that we were better positioned with the ball so we had the ball a little bit more. In the second half, we just kept running.
On moving top of the league:
"It's more if you play against Arsenal in their stadium and you can get a draw away from home. Brighton are a very strong team as well, they have shown this season. If you face a team like this and are 1-0 down and can come back to a 2-1 win which in my opinion was deserved that gives me a lot of confidence. More than the position we are in."
Brighton boss Fabio Hurzeler:
"I'm disappointed. I think my team deserved more from the result but in the end, it's Liverpool with their individual quality. It was our responsibility in the second half that we didn't carry on playing like in the first half. We expected them to press more but there were still solutions, in the end, we didn't manage that. Their fouls were outside of the box, with luck, and sometimes that's football.
"All the 50/50 moments went to them and also the 50/50 duels the referee gave to them. I don't understand why, to be honest, maybe the atmosphere was too loud. Momentum went to their side and then they made their goals. Bad, bad experience."