Mohammed Salah has now scored 34 goals for Liverpool in the Champions LeagueJoel Matip's late header against Ajax at Anfield gave Liverpool their first Champions League win this season.
Boss Jurgen Klopp demanded a reaction to the "horror show" of the 4-1 defeat at Napoli in their group opener and this was certainly a vast improvement.
Liverpool created plenty of openings and deservedly went ahead when Mohamed Salah swept home Diogo Jota's pass.
Mohammed Kudus' searing shot went in off the underside of the bar to level but Matip's 89th minute header won it.
Daley Blind missed a great chance to put Ajax ahead late on before Matip - who last scored in the Champions League nine years ago - rose to head the winner, which was confirmed by goal-line technology.
Liverpool's persistence pays off
Liverpool needed to respond to the debacle of Napoli and while they were nowhere near their best this was all about getting a victory that would make their Champions League group look more positive.
It looked for a long time that Ajax's composed, unhurried style, allied to Liverpool's lack of cutting edge, might mean another disappointment before Matip rose to score.
It meant a vital three points for last season's beaten finalists and the start of their recovery from the opening reverse in Italy that Klopp labelled the worst performance of his reign at Anfield.
Before the game, there was a short period of silence in tribute to Queen Elizabeth II and while there were a handful of dissenting voices shouting out, they were hushed by the vast majority of those inside Anfield, who observed the tribute respectfully.
Once the action started, match-winner Matip and Thiago Alcantara were the key to Liverpool's improvement, the defender not only being strong at the back but bringing the ball out of defence in his typical manner.
The former Schalke defender's decisive late contribution capped an outstanding display and made Klopp's defence look far more solid than in Italy last week.
Thiago's ability to dictate the tempo has been sorely missed while he has been injured and his performance was a signpost to how important he will be as Liverpool try to recover from an indifferent start to their season.
Kudus shows his class
Ajax are surpassed by very few clubs in world football in terms of unearthing talent and the latest gem looks to be Ghana forward Kudus.
The 22-year-old forward was strongly linked with a move to Everton at the end of the transfer window but Ajax blocked the deal given they were about to lose Antony to Manchester United.
Ajax believe Kudus will make a key contribution to their season - and presumably that his value will therefore significantly increase.
So it has proved. He was outstanding in the 4-0 thrashing of Rangers in Ajax's opening group game and his fierce finish at Anfield was a sign of the quality the powerful, skilful forward possesses.
He thumped in an emphatic finish off the underside of the bar after he was played in by Steven Berghuis and Steven Bergwijn.
Ajax will be naturally be disappointed with the final outcome given they were minutes away from a valuable point, but they will justifiably feel they showed enough to believe they can still progress from what is shaping up as a tight group.
Player of the match
MatipJoël Matip
Ajax
Squad number22Player namePasveer
Squad number20Player nameKudus
Squad number3Player nameBassey
Squad number15Player nameRensch
Squad number4Player nameÁlvarez
Squad number2Player nameTimber
Squad number7Player nameBergwijn
Squad number17Player nameBlind
Squad number8Player nameTaylor
Squad number23Player nameBerghuis
Squad number10Player nameTadic
Squad number9Player nameBrobbey
Squad number19Player nameSánchez
Squad number21Player nameGrillitsch
Line-ups
Liverpool
Formation 4-3-3
1Alisson
66Alexander-Arnold32Matip4van Dijk21Tsimikas
19Elliott3Fabinho6Thiago Alcántara
11Salah20Jota23Díaz
1Alisson66Alexander-Arnold32MatipBooked at 62mins4van Dijk21Tsimikas19ElliottSubstituted forFirminoat 66'minutes3Fabinho6Thiago AlcántaraSubstituted forBajceticat 90+4'minutes11Salah20JotaSubstituted forNúñezat 66'minutes23DíazSubstituted forMilnerat 90+2'minutesSubstitutes
2Gomez7Milner9Firmino13Adrián27Núñez28Freitas Gouveia Carvalho29Arthur43Bajcetic47Phillips95DaviesAjax
Formation 4-2-3-1
22Pasveer
15Rensch2Timber3Bassey17Blind
4Álvarez8Taylor
10Tadic23Berghuis7Bergwijn
20Kudus
22Pasveer15RenschSubstituted forSánchezat 68'minutes2Timber3Bassey17Blind4ÁlvarezBooked at 59mins8TaylorSubstituted forGrillitschat 80'minutes10Tadic23BerghuisBooked at 90mins7Bergwijn20KudusSubstituted forBrobbeyat 86'minutesSubstitutes
1Stekelenburg6Klaassen9Brobbey11Ocampos13Kaplan16Gorter18Lucca19Sánchez21Grillitsch25Baas26Regeer29MagallánReferee:Artur Manuel Soares DiasAttendance:52,387Live Text
Match ends, Liverpool 2, Ajax 1.
Second Half ends, Liverpool 2, Ajax 1.
Substitution, Liverpool. Stefan Bajcetic replaces Thiago.
Hand ball by Kostas Tsimikas (Liverpool).
Substitution, Liverpool. James Milner replaces Luis Díaz.
Steven Berghuis (Ajax) is shown the yellow card.
Luis Díaz (Liverpool) wins a free kick in the defensive half.
Foul by Steven Berghuis (Ajax).
Goal! Liverpool 2, Ajax 1. Joël Matip (Liverpool) header from the centre of the box to the high centre of the goal. Assisted by Kostas Tsimikas with a cross following a corner.
Corner, Liverpool. Conceded by Calvin Bassey.
Attempt blocked. Mohamed Salah (Liverpool) left footed shot from the centre of the box is blocked.
Substitution, Ajax. Brian Brobbey replaces Mohammed Kudus.
Foul by Thiago (Liverpool).
Steven Berghuis (Ajax) wins a free kick in the defensive half.
Corner, Liverpool. Conceded by Remko Pasveer.
Attempt saved. Darwin Núñez (Liverpool) right footed shot from a difficult angle on the left is saved in the bottom right corner. Assisted by Roberto Firmino.
Attempt missed. Darwin Núñez (Liverpool) right footed shot from the centre of the box is close, but misses to the left. Assisted by Mohamed Salah.
Joël Matip (Liverpool) wins a free kick on the right wing.
Foul by Steven Bergwijn (Ajax).
Substitution, Ajax. Florian Grillitsch replaces Kenneth Taylor.
