Atsu levelled proceedings for Rafael Benitez's side seven minutes after Virgil van Dijk opened the scoring for the visitors in the 13th minute.
The effort was the Ghana midfielder's maiden goal in the Premier League this season after 27 league outings.
Salah later restored the Reds' lead in the 28th minute with his 22nd league goal of the season.
After the restart, Salomon Rondon's 54th-minute equaliser was not enough to secure a point for Newcastle at home as Divock Origi came off the bench to net the match-winning goal four minutes from time.
Liverpool moved back to the summit of the Premier League table with 94 points from 37 matches, but they were dealt an injury blow after talisman Salah was stretchered off in the 73rd minute.
Jurgen Klopp's side will hope to have the Egyptian winger back in time for their final Premier League game of the season against Wolverhampton Wanderers on May 12, while Atsu's Newcastle United will aim to bounce back from the loss against relegated Fulham on the same day.