Jurgen Klopp's squad fell behind early on against their struggling opponents as Jonjo Shelvey scored against his previous club following a rapid break, despite the absence of Virgil van Dijk and Fabinho due to suspected positive coronavirus tests.
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Liverpool, on the other hand, took command after scoring two goals in the span of four minutes.
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Diogo Jota's equaliser was controversial since it occurred when Isaac Hayden was down in the six-yard box with a head injury, but the Portuguese took advantage of the situation by tucking in on the rebound after Martin Dubravka's first save.
Allan Saint-Maximin missed a chance to tie Jamie Vardy's record of scoring or assisting in 15 consecutive Premier League games just before Salah's goal.
After the interval, Eddie Howe's side improved as the hosts relaxed, and went near via Jacob Murphy and Shelvey, but Trent Alexander-late Arnold's effort from 25 yards finished the game, and Liverpool are now a point behind Manchester City.
Newcastle is still in 19th place in the table, three points from safety, and their mission of avoiding relegation will be made even more difficult when they face Pep Guardiola's league leaders on Sunday.