Jurgen Klopps men left it late to seal an important Champions League victory, with Joel Matip the unlikely hero late in the day.
The three points restored some parity in Group A, as Liverpool were much-improved after the embarrassment at Napoli.
Heres how the media reacted to the Reds thrilling win.
Richard Jolly of the Independent compared Matips goal to Dejan Lovrens against Borussia Dortmund:
Arguably Jurgen Klopps first great European night at Anfield culminated in late winner from a centre-back sent up for a set-piece.
If Ajax in 2022 may not yet belong alongside Dortmund in 2016, Joel Matips decider here might rank alongside Dejan Lovrens against Klopps old employers six years ago in terms of its importance.
After the worst display of his reign and the nightmare in Naples, Liverpool required a response and a win.
In their energy and their organisation, in the individuals and the collective, they had improvement. But as long as Ajax were on course to take a point, there was a case for arguing the result could be still more damaging than their statutory setback in Naplesuntil Matip met Kostas Tsimikas corner to plant a header beyond Remko Pasveer and frustration was replaced by celebration.
The Liverpool Echos Paul Gorst saw clear signs of improvement:
After administering four or five days of absolute truth to his Liverpool players last week, it seems Jurgen Klopp has found that honesty is indeed the best policy.
This was not exactly a Reds side at their swashbuckling, awe-inspiring best, but they at least started to vaguely resemble the team who were good enough to reach the final of the Champions League just a few short months ago.
For a sport that does hyperbole and overreaction so well those sorts of statements were always going to be forthcoming, but the only way to turn down the volume on that noise was by a proper Liverpool reaction here against Ajax, one befitting of their well-earned reputation.
This was perhaps not that, but a welcome step in the right direction as Klopps men signed off for September with a much-needed return to winning ways.
Sam Wallace of the Telegraph was another felt it was much better from Liverpool, although there is still work to be done:
Jurgen Klopp may not yet be able to herald the return of the form that made some of his players giants of Liverpools recent past, although at least this time when it mattered most there was something that resembled the best performances of the Germans era.
In years past they might have bulldozed through Ajax much sooner than the 89th minute but something remains off-key about this Liverpool side.
They are not quite capable of crushing opposition the way they once were.
Jolly added that he thought Klopp got his team selection spot on at Anfield:
Vindication for Klopp, perhaps. He had brought four players into the side. Each played his part. Matip scored, Tsimikas and Diogo Jota earned assists and Thiago Alcantara lent class in possession.
Just as pertinently, each played well. So did Liverpool, albeit imperfectly. On a day when they had 23 shots, they eventually mustered a second goal to ensure shortcomings in either box did not cost them.
Klopp could find no positives to derive from the show of incompetence in Naples. There were several here, including the sight of Mohamed Salah on the scoresheet.
Goals Neil Jones liked what he saw from the goalscoring Mohamed Salah:
You cant keep the worlds best players down for long. Salahs start to the season has, like Liverpools, been a mixed one, but this was much more like it from the Egyptian.
He buried the first chance that came his way, latching onto Diogo Jotas pass and finishing sharply on his left foot after 17 minutes.
It was exactly what Jurgen Klopp will have wanted to see, his forwards linking up to devastating effect, with Salah profiting centrally instead of being marooned on the right flank as in recent weeks.
All in all, a welcome step in the right direction for the Reds star man.
The Daily Mails Dominic King felt Darwin Nunez really struggled, though:
This is not making a snap judgment or suggesting that Liverpool have bought a dud in Nunez. The quality of clubs who wanted to take him from Benfica this summer and who tried in January tells you he is a young man with talent and, in flashes, he has shown it in red.
But the very fact Klopp has used Nunez in bit-part roles, as Liverpool set off on a road they hope will lead them back to Istanbul next June, tells you that his acclimatisation is taking longer than some would have anticipated.
Perhaps the early cameos against Manchester City in the Community Shield, when he won a penalty and then scored the decisive third goal in a 3-1 victory, and against Fulham, when he scored again and set up another for Salah have not been particularly helpful.
After the City triumph, the narrative became who would score more Nunez or Erling Haaland but the Norwegians staggeringly smooth transition has made this a non-contest. How can it be when he is only fleetingly on the pitch?
The Mirrors Daniel Orme was critical of another iffy Trent Alexander-Arnold showing:
Whilst great acclaim has to go to Kudus for his goal, Liverpool defender Trent Alexander-Arnold might reflect on an evening where his defensive contributions have been called into question once more.
The Liverpool right-back allowed Steven Berghuis to ghost in-behind him and pull the ball back towards Steven Bergwijn. The ex-Spurs man missed his effort, only to see it fall to Kudus.
On what was his 50th appearance for the Reds in Europe, the right-backs defensive awareness was left wanting once and it is becoming ever clearer it is something he certainly needs to work on.
Finally, This Is Anfields Mark Delgado hailed the returning Matip and Thiago:
First and foremost, the return of Thiago was as glorious and necessary as youd expect, with his patented body swerves and excellent passing in the centre all on show in the first few minutes.
Hes a class above more often than not; the only problem is that he was a class above against his own team too, with few able to match his incision on the ball.
Second, Joel Matip was very good, on and off the ball. He made a sneaky foul or two when he had to, he intercepted and made headed clearances but he also brought the ball of defence with great skill and poise, picking a good pass at the end of it.