The Reds faced their most bitter rivals, looking to pull further clear at the top of the table after slip-ups by Man City and Leicester.
Fabinho made a welcome return to the matchday squad, having not featured since the 1-1 draw with Napoli back in November, making a late cameo appearance.
Roberto Firmino thought he had added a second for the hosts less than 10 minutes later, but Van Dijk was harshly adjudged to have fouled David de Gea.
Liverpool were dominant, with a dramatic sunset on a misty day on Merseyside providing a spectacular backdrop at Anfield.
Captain Jordan Henderson continued his imperious form with a superb performance, showing great quality and leadership throughout.
In a tense finale, Klopp grew increasingly edgy, as did the entire Anfield crowd, with United threatening to score an equaliser.
It never arrived.
Alisson was the first player to celebrate with his teammate, bringing back echoes of Pepe Reina and David N’Gog against United back in 2009.
After the final whistle, Henderson and Salah basked in the glory of victory after the game, having been interviewed on the pitch.