However, Divock Origi came off the bench to score a crucial 86th-minute winner which sees Liverpool move back above Manchester City at the top of the table until at least Monday, when the champions host Leicester City in their penultimate game of the campaign.
Liverpool were once again without Roberto Firmino due to injury, but that did not prevent them from making a quick start to the match, and Salah snatched at an early effort after the ball had dropped to him inside the box.
The visitors would not have to wait long for the opening goal, though, and it was no surprise to see it come via the aerial route with Liverpool having scored more goals from headers than any other side this season, while no team has conceded more of them than Newcastle.
The Magpies did not do themselves any favours, though, allowing Van Dijk to meet the ball unmarked on the edge of the six-yard box and plant a straightforward header past a helpless Martin Dubravka.
It was an ideal start in a tricky fixture for Liverpool, but the lead lasted only seven minutes before Newcastle restored parity in dramatic and controversial fashion.
Matt Ritchie dragged his shot into the path of Rondon, who looked certain to score from point-blank range until Trent Alexander-Arnold purposefully stuck out his elbow to deny a guaranteed goal on the line.
However, the England international - back in the starting XI after missing out against Barcelona in midweek - was spared a red card and a penalty because Atsu stroked home the rebound, which was perversely the best-case scenario for Liverpool despite giving up their lead.
The equaliser did spark Newcastle into life, though, and Javi Manquillo drew a save from Alisson with a diving header before Ayoze Perez crashed a snapshot against the crossbar moments later after picking up Atsu's knockdown.
Liverpool regained the lead just as Newcastle were enjoying their best spell of the first half, though, and it was Alexander-Arnold who made the most of his disciplinary reprieve to provide his record-equalling 11th Premier League assist of the season.
The full-back swung a cross into the box for Salah, who had peeled away from his marker and simply steered a controlled right-footed volley past Dubravka into the bottom corner.
Liverpool should have added a third shortly after the half-hour mark when Salah threaded a perfectly-weighted ball through for Sadio Mane, but Dubravka was quick off his line to thwart the in-form Liverpool forward.
Daniel Sturridge was the next to come close in an action-packed first half when he curled a shot a couple of yards past the top corner, but Newcastle were still posing a threat at the other end and Rondon drew a routine stop from Alisson shortly before half time.
It was a reminder to Liverpool that a one-goal lead was not safe, and Rondon in particular was causing problems for the Liverpool defence - something he continued to do in the second half with a sharp turn and shot which forced another save five minutes after the restart.
Liverpool could have given themselves the cushion they needed seconds later when Georginio Wijnaldum teed up Sturridge on the edge of the box, only for the striker to blaze his finish over the crossbar when he should have at least hit the target.
It looked like being a costly miss when Newcastle levelled things up again nine minutes into the second half, with Rondon lashing a left-footed volley into the bottom corner after Liverpool had failed to properly clear their lines from a corner.
There was more bad news for the visitors 20 minutes from time when Salah suffered a heavy blow to the head from Dubravka before also hitting his head against the ground when falling, leading to a lengthy stoppage and the Premier League top-scorer being carried off the field on a stretcher in worrying scenes.
Salah was replaced by Origi, and the Belgian went on to have the all-important say when he got his head to Xherdan Shaqiri's free kick, although the final touch appeared to come off Jamaal Lascelles as it flew over the line.
It was a goal which kept Liverpool's title hopes alive just when it looked as though they might be fading, and the Reds successfully saw out eight minutes of stoppage time to put the pressure back on Man City ahead of Monday's match against Leicester.
An eighth successive league win for Liverpool takes them on to 94 points - already the third-highest total in Premier League history - with one game remaining, while also equalling their club record of 13 away league wins in a season.
NEWCASTLE (5-4-1): Dubravka; Manquillo, Schar (Muto 91'), Lascelles, Dummett, Ritchie; Perez, Hayden, Ki, Atsu; Rondon
LIVERPOOL (4-3-3): Alisson; Alexander-Arnold, Lovren (Milner 83'), Van Dijk, Robertson; Henderson, Fabinho, Wijnaldum (Shaqiri 66'); Salah (Origi 73'), Sturridge, Mane