Curtis Jones took advantage of a huge error from Ajax goalkeeper Andre Onana to score the only goal, while Caoimhin Kelleher kept a clean sheet against the Dutch giants on his European debut.
Victory also ensures Jurgen Klopp's men win Group D with a game to spare, allowing the German to give some of his stars a much-needed rest away to Midtjylland next week.
Kelleher had to deputise for Alisson Becker, who added to Klopp's long injury list before kick-off, while Andrew Robertson needed heavy strapping applied to his foot in the first half.
Liverpool were already without Virgil van Dijk, Joe Gomez, Alex Oxlade-Chamberlain, Naby Keita, Xherdan Shaqiri, Trent Alexander-Arnold and James Milner through injury.
Jones has been one of the beneficiaries of those absentees with many more first-team minutes this season and the 19-year-old came closest to breaking the deadlock before half-time when his curling effort came back off the post after just six minutes.
Ajax had been unbeaten in eight games since losing to at home to Liverpool in October and the visitors soon settled into their rhythm to dominate possession.
All of Kelleher's previous appearances had come in domestic cup competitions and the young Irishman was made to work for his clean sheet on the big stage as he flew to his left to turn Noussair Mazraoui's long-range effort behind.
Mazraoui was denied again by Kelleher early in the second half, but David Neres should have swept home the rebound when he hit the outside of the post.
Ajax now must beat Atalanta in Amsterdam next week to make the last 16 and they were left to regret what might have been but for one moment of madness from Onana.
A lofted cross from Neco Williams appeared easy pickings for the Ajax 'keeper but he instead tried to let the ball go behind for a goal kick and was caught out when Jones sneaked in at the back post to turn the ball into an unguarded net.
The Spaniard made amends with a great save to turn Roberto Firmino's effort onto the post to keep Ajax in the game after Liverpool's best move of the match involving Jordan Henderson and Salah.
But Kelleher produced an even better stop two minutes from time to parry Klaas-Jan Huntelaar's powerful header from point-blank range to see Liverpool over the line.
A delighted Klopp ran onto the field to embrace his goalkeeper after the full-time whistle and the 2018/19 Champions League winners will hope to be in better health for another deep run in the knockout stages come the new year.