There were 30,364 fans present to witness a sublime away display helped by Roberto Firmino's double.
Emre Can opened the scoring on the half-hour with a header from Philippe Coutinho’s corner.
Within 80 seconds Coutinho’s cross was angled in by Firmino from six yards.
Mohamed Salah teed up Firmino on the counter-attack for the Brazilian striker to find the top corner on 48 minutes.
Glenn Murray dispatched a penalty for the hosts three minutes later, but Jurgen Klopp’s men clinched three points thanks to Coutinho's free-kick and Lewis Dunk’s own goal.
Liverpool climb into the top four, with Arsenal losing to Manchester United in the late kick-off. Brighton drop to 11th.
Brighton: 9 Dec v Huddersfield (A)
Liverpool: 10 Dec v Everton (H)