Jurgen Klopp's men had fallen behind in the seventh minute when Jetro Willems cut inside and rifled home to shock the Anfield crowd.
Wolves' Patrick Cutrone further reduced the deficit late on, only for another Chelsea youngster, Mason Mount, to wrap up the win in stoppage time on an afternoon that belonged to one man.
Brighton are now 15 in the Premier League with five points from five games. Away teams hoping to leave Merseyside with three points are facing football's equivalent of trying to escape from Alcatraz. "I am happy", Salah was quoted as saying by Liverpool Echo.
"You have to look at the two boys he's got either side of him, 20-plus goals each last season?" Nowadays in the Premier League every opponent is hard so we knew that it was going to be tough. They will sharpen the note when entertaining Newcastle at Anfield.
What was most striking about this success, one that unexpectedly saw Liverpool five points clear by the close of play, was the ease with which they went through the gears. They've scored a league-low three in their opening four games.
Paul Pogba heads a lengthy United injury list for the visit of in-form Leicester with Solskjaer potentially missing seven first-team players. It was exactly what they did and once Sadio Mane scored a fabulous equaliser there was an inevitability about the outcome.
Jurgen Klopp made two changes to the side that beat Burnley before the worldwide break as Alex Oxlade-Chamberlain and Divock Origi replaced Jordan Henderson and Roberto Firmino. We know that. But we have to be from time to time sensible and leave him out.
Even after this goal, Liverpool kept searching for a fourth but Firmino was offside in the buildup to a chance that he created for Sadio Mane to complete his hattrick.
For those paying attention, Salah celebrated with a punch of the air.
Tottenham director Mauricio Pochettino expected that the conclusion of this European transfer window could settle his squad along with the Champions League finalists appeared more like their older selves by ripping apart Palace by the away. They are only intent on the future.
"We hung in there", United manager Ole Gunnar Solskjaer said.
The Brazilian bagged another assist when his flick freed Salah, who took his time before leaving Dubravka grounded and placing the ball into the far corner.
On Twitter, Neville took part in a Q&A and was asked if he thought Liverpool would win the league this year.
"And now my focus is on recovery and being ready for Newcastle on Saturday". Now we focus on Napoli.
He said: "We have acquitted ourselves very well up to a certain level".
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