Salah had to be substituted following his goal with a foot injury but that was the only blip on an enjoyable afternoon for Jurgen Klopp's side.
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Liverpool are now three points behind City and face a challenging trip to fourth-place chasing Arsenal on Wednesday.
"We were good, we knew Brighton are a really good team that can create problems, that's what we saw from the beginning," Klopp told BT Sport in a postmatch interview.
"Like the last game we need to get used to the opponents. They adapt to us a lot and play in areas to try and hurt us. We were really strong and scored a great goal.
"We had really good moments with massive chances where we missed the final pass. But I'm happy. There were moments at the the end where I'd like to control the ball more but it's a difficult moment."
Brighton started the game as the better team but it was Liverpool who took the lead after 19 minutes.
VAR took a look at the incident for violent conduct but the goalkeeper did not receive a card for the late and high challenge.
The visitors started to control the game after the goal and had a chance to double their lead before the break but Salah's tame effort was easily saved by Sanchez.
Brighton should have equalised a minute into the second half but Leandro Trossard blazed over the bar from close range.
Salah almost made Brighton pay moments later but his deflected effort came off the crossbar and then missed another good chance minutes later.
Salah smashed the ball down the middle to double Liverpool's lead and score the club's 2,000th goal in Premier League history -- only Manchester United have scored more goals in the competition.
The forward was taken off shortly afterwards with a knock to his foot and was replaced by Diogo Jota.
"It is always pleasing to come away from home and get the three points," Liverpool captain Jordan Henderson said after the game. "Brighton are a good footballing side. I know they've lost the last few but they are always dangerous.
"They've troubled us at times over the past couple of years so it was a really big game for us today and we delivered. We are delighted."