Allison saved his first Premier League penalty for Liverpool since September 2020 against ChelseaDarwin Nunez scored his first Anfield goal for Liverpool and Alisson saved a penalty as the Reds defeated West Ham in an incident-packed encounter.
Nunez had already gone close with a rasping attempt when he rose to meet Kostas Tsimikas' cross and plant a downward header beyond the reach of Lukasz Fabianski.
The busy Nunez was presented with a chance to double the lead moments later but this time Fabianski was able to save before the Hammers were awarded a penalty after Joe Gomez's foul on Jarrod Bowen.
Bowen picked himself up to take it but Alisson produced a fine diving save in front of the Kop to ensure Liverpool backed up their excellent win over Manchester City on Sunday with another three points.
Nunez starting to settle
Liverpool fans turned up at Anfield wondering whether the team could follow up the win over Manchester City on Sunday.
The Reds have been unpredictable this season but they made it three straight wins in all competitions as Nunez made his mark.
It has not been an easy start to life for the Uruguay striker who arrived to much fanfare from Benfica for an initial £64m in the summer.
Against West Ham, Nunez offered further evidence he is starting to find his rhythm and confidence.
He put himself in an excellent position to meet Tsimikas' cross and beat Fabianski for his third goal in four games.
Nunez was lively throughout and could have had a hat-trick but 37-year-old Fabianski denied him with two fine saves while the Liverpool forward also hit the post with a powerful attempt.
His scoring streak is welcome news for manager Jurgen Klopp who has lost Diogo Jota and Luis Diaz to injury for the foreseeable future.
Hammers pay the price
Nobody could seriously question the job David Moyes has done since returning to West Ham in December 2019.
Two top-seven finishes and a Europa League semi-final is evidence of a club moving forward but the Hammers are struggling for consistency after a sixth defeat in 11 Premier League games.
Moyes said the assistant referee "needs to go to Specsavers" after being angered by the officials during the 1-1 draw at Southampton on Sunday.
But his side only had themselves to blame for not taking their chances at Anfield.
Referee Andre Marriner awarded the Hammers a penalty shortly before half-time after checking the pitchside monitor following Gomez's foul on Bowen.
But the England forward was brilliantly denied by Alisson, who followed up his assist for Mohamed Salah's winner against Manchester City with a crucial save.
Substitute Said Benrahma and Gianluca Scamacca also had great chances to equalise while, in the closing moments, James Milner produced a goal-saving challenge to deny Tomas Soucek a certain equaliser.
The Hammers have now scored just nine goals in 11 Premier League games and their lack of spark in front of goal will worry Moyes.
Line-ups
Liverpool
Formation 4-4-2
1Alisson
66Alexander-Arnold2Gomez4van Dijk21Tsimikas
11Salah14Henderson6Thiago Alcántara28Freitas Gouveia Carvalho
9Firmino27Núñez
1Alisson66Alexander-Arnold2Gomez4van Dijk21TsimikasSubstituted forRobertsonat 81'minutes11Salah14Henderson6Thiago AlcántaraSubstituted forFabinhoat 58'minutes28Freitas Gouveia CarvalhoSubstituted forJonesat 57'minutes9FirminoSubstituted forMilnerat 81'minutes27NúñezSubstituted forElliottat 57'minutesSubstitutes
3Fabinho7Milner15Oxlade-Chamberlain17Jones19Elliott26Robertson42Clark47Phillips62KelleherWest Ham
Formation 4-4-2
1Fabianski
2Johnson4Zouma24Kehrer3Cresswell
12Downes28Soucek41Rice8Fornals
20Bowen7Scamacca
1Fabianski2Johnson4Zouma24Kehrer3Cresswell12DownesSubstituted forAntonioat 74'minutes28Soucek41Rice8FornalsSubstituted forBenrahmaat 61'minutes20Bowen7ScamaccaSubstitutes
5Coufal9Antonio10Lanzini13Areola21Ogbonna22Benrahma32Coventry33Emerson35RandolphLive Text
Match ends, Liverpool 1, West Ham United 0.
Second Half ends, Liverpool 1, West Ham United 0.
Harvey Elliott (Liverpool) wins a free kick in the defensive half.
Foul by Aaron Cresswell (West Ham United).
Corner, Liverpool. Conceded by Declan Rice.
Attempt blocked. Mohamed Salah (Liverpool) left footed shot from the centre of the box is blocked. Assisted by James Milner.
Attempt blocked. Jordan Henderson (Liverpool) right footed shot from outside the box is blocked. Assisted by James Milner.
Attempt missed. Gianluca Scamacca (West Ham United) header from the centre of the box misses to the right. Assisted by Aaron Cresswell with a cross following a corner.
Corner, West Ham United. Conceded by Alisson.
Attempt saved. Tomás Soucek (West Ham United) right footed shot from very close range is saved in the bottom left corner. Assisted by Jarrod Bowen.
Foul by James Milner (Liverpool).
Ben Johnson (West Ham United) wins a free kick in the defensive half.
Attempt saved. Mohamed Salah (Liverpool) left footed shot from the right side of the box is saved in the bottom right corner. Assisted by Trent Alexander-Arnold.
Substitution, Liverpool. James Milner replaces Roberto Firmino.
Substitution, Liverpool. Andrew Robertson replaces Konstantinos Tsimikas.
Attempt missed. Gianluca Scamacca (West Ham United) right footed shot from the right side of the box is close, but misses the top right corner. Assisted by Michail Antonio following a fast break.
Corner, Liverpool. Conceded by Kurt Zouma.
Offside, Liverpool. Trent Alexander-Arnold tries a through ball, but Mohamed Salah is caught offside.
Hand ball by Michail Antonio (West Ham United).
Corner, Liverpool. Conceded by Declan Rice.
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