Liverpool move onto 40 points from 17 games, one behind leaders Manchester City who beat Leeds United 7-0 on Tuesday.
With Virgil Van Dijk, Fabinho and Curtis Jones ruled out for the hosts, Jonjo Shelvey struck the opener for Newcastle against the run of play in the seventh minute, with a swerving, dipping drive past unsighted keeper Allison Becker at Anfield.
The lead lasted 14 minutes before Diogo Jota fired home the rebound after Martin Dubravka saved his initial header to level, with Newcastle's players complaining the ball should have been put out of play as Isaac Hayden was down injured in his own box.
Mohamed Salah put Liverpool ahead four minutes later, equalling Leicester City striker Jamie Vardy's record of 15 consecutive Premier League games with a goal or assist by blasting home the rebound form a Sadio Mane shot.
Newcastle improved dramatically in the second half, showing a defensive resolve that has been absent for much of the season, but in the last 10 minutes Liverpool started to turn the screw.
Dubravka had to get down smartly to save from Mane in the 81st minute and Jota had a close-range shot blocked before the Newcastle keeper pulled off another super stop from Naby Keita.
There was nothing he could do three minutes from the end of normal time though as Alexander-Arnold took a touch on the edge of the box before firing a rocket into the top corner.
"I've been waiting for that for five years, waiting to hit it clean like that," Alexander-Arnold told BT Sport. "I've had a few from the edge of the box ... I've caught that one sweet and it's just nestled in the top corner."
The England fullback said he and his teammates had done their best to ignore the possible positive Covid tests at the club as the games come thick and fast over Christmas.
"We're focused on our football until we're told otherwise. We'll be in there every day, recovery again tomorrow and make sure we're fit for Sunday," he said.
"We've got a big game, going down to Tottenham - it'll be an interesting one but all we can do is focus on that and prepare for the game."