The Premier League champions were unconvincing and struggled to find their rhythm in their first game without Virgil van Dijk after the central defender suffered a potentially season-ending injury in the Merseyside derby draw against Everton on Saturday.
Fabinho, who has usually operated in midfield for Liverpool, had an outstanding game, notably hooking a clearance off the line to deny Dusan Tadic just before the break.
It was one of several chances for Ajax in the first half, as Lisandro Martinez had a header saved by Adrian and Ryan Gravenberch shot narrowly wide, much to the annoyance of Liverpool manager Klopp, who spent the early part of the game berating and shouting instructions to his players.
The visitors took the lead on 35 minutes when Sadio Mane turned and dribbled into the box and scuffed his shot at Tagliafico who deflected the effort into his own net.
Ajax made a quick start to the second-half with Davy Klaassen hitting the inside of the post with a curling shot from the edge of the penalty area.
Both sides traded blows with Fabinho's header deflected wide at one end, while at the other Adrian saved to deny Quincy Promes.
Mane was seen with ice on his leg after coming off and when asked about it, Klopp said: "For a few days he's had a dead leg -- in that area -- it's still painful, and it's not a problem, when he doesn't play, he puts ice on it."
The substitutes made a good impression, Jota cutting in from the left and forcing a corner with a shot as well as chasing down Andre Onana in the Ajax goal and forcing an error with the goalkeeper off his line.
Minamino came close to setting up a second when he fired the ball across the face of goal but Andy Robertson was unable to slide in to score from close-range.
Ajax threw their players forward at the end and should have equalised four minutes into stoppage time when Adrian did not properly deal with a cross into the area but substitute Jurgen Ekkelenkamp missed the target.