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By Michael EmonsBBC Sport at Anfield
Bottom-of-the-table Blades were on course for a shock draw at Anfield, but Mac Allister - a World Cup winner with Argentina - scored with a brilliant 20-yard strike into the top corner to put the Reds 2-1 up.
Cody Gakpo added a 90th-minute third as Liverpool returned to the top of the table.
Boss Klopp said: "We needed Macca's wonder goal to turn it around.
He is super important for us, a wonderful player and a wonderful boy - I'm really happy for Liverpool that we got him.
"His defensive work has been really good where people were doubting if he could be defensive, but I prefer him in the eight [attacking midfield].
"He is a really good player.
Then the free-kick directly after [the goal] - absolutely insane, what a player."
Mac Allister, who joined Liverpool for £35m from Brighton last summer, also hit the crossbar with a curling free-kick.
Liverpool were gifted a 17th-minute lead when Darwin Nunez charged down goalkeeper Ivo Grbic's clearance with the ball rebounding into the net.
Despite having 84% of possession in the first half and wasting numerous chances, the hosts conceded a 58th-minute equaliser as Gustavo Hamer's header bounced off Conor Bradley and into his own net.
But Mac Allister - nominated for March's Premier League Player of the Month award - put the hosts back in front with 14 minutes left, before substitute Gakpo's header sealed a vital success.
The result moved the Reds two points clear of Arsenal and three ahead of reigning champions Manchester City, with eight league games left.
More importantly, it also kept Liverpool's title destiny in their own hands.
They know if they beat Manchester United, Crystal Palace, Fulham, Everton, West Ham, Tottenham, Aston Villa and Wolves, it will secure their second Premier League title and confirm them as top-flight English champions for a 20th time.
Reds rally for victory, but late goals frustrate Wilder
The visitors were looking to win at Anfield for the first time since Jostein Flo scored twice in a 2-1 win on 2 April, 1994, while Liverpool had not lost a home league match in 27 games - a 2-1 reverse against Leeds in October 2022.
But Sheffield United nearly made a dream start, only for home goalkeeper Caoimhin Kelleher to make an excellent save to deny James McAtee in the opening 90 seconds.
Then, against the run of play, the Blades equalised as Anfield started to get nervous in what could have been a devastating loss of two points.
It needed something special to change the course of the outcome - and Mac Allister provided it with a glorious strike for one of the goals of the season to calm the home fans' nerves before Gakpo put the game to bed.
For the struggling Blades, they would not have expected to get anything from Anfield, with boss Chris Wilder admitting beforehand they needed "a huge night" if they were to cause an upset.
"The narrative at the press conference before the game and in the interviews was a sense that we were just here to make the numbers up and this was going to be a comfortable evening for Liverpool," said Wilder afterwards.
"That was not our motivation, or our emotions and we try to do everything for the club.
In the last 15 minutes, you saw the power of the Premier League, and our Achilles heel has been conceding late goals."
The Blades are trying to avoid an instant return to the Championship and are 10 points adrift of safety, although they still have four of their relegation rivals - Brentford, Burnley, Nottingham Forest and Everton - still to play.
Liverpool, Premier League winners under Klopp in 2019-20, next face a huge game on Sunday at Manchester United - the side that beat them 4-3 after extra time in a thrilling FA Cup quarter-final at Old Trafford last month.
LiverpoolLiverpoolLiverpoolSheff UtdSheffield UnitedSheffield UnitedLiverpoolSquad number10Player nameMac AllisterAverage rating
7.43
Squad number9Player nameNúñezAverage rating
7.30
Squad number18Player nameGakpoAverage rating
7.27
Squad number4Player namevan DijkAverage rating
7.10
Squad number7Player nameDíazAverage rating
7.05
Squad number26Player nameRobertsonAverage rating
7.01
Squad number19Player nameElliottAverage rating
6.98
Squad number17Player nameJonesAverage rating
6.96
Squad number62Player nameKelleherAverage rating
6.86
Squad number84Player nameBradleyAverage rating
6.71
Squad number8Player nameSzoboszlaiAverage rating
6.68
Squad number5Player nameKonatéAverage rating
6.65
Squad number2Player nameGomezAverage rating
6.65
Squad number11Player nameMohamed SalahAverage rating
6.54
Squad number38Player nameGravenberchAverage rating
6.30
Sheffield United
Squad number11Player nameBrereton
Average rating
5.95
Squad number8Player nameHamer
Average rating
5.93
Squad number28Player nameMcAtee
Average rating
5.84
Squad number24Player nameArblaster
Average rating
5.78
Squad number23Player nameOsborn
Average rating
5.71
Squad number9Player nameMcBurnie
Average rating
5.65
Squad number32Player nameOsula
Average rating
5.62
Squad number20Player nameBogle
Average rating
5.61
Squad number5Player nameTrusty
Average rating
5.59
Squad number21Player nameVinicius Souza
Average rating
5.54
Squad number15Player nameAhmedhodzic
Average rating
5.54
Squad number30Player nameHolgate
Average rating
5.49
Squad number19Player nameRobinson
Average rating
5.49
Squad number35Player nameBrooks
Average rating
5.35
Squad number13Player nameGrbic
Average rating
5.19
Squad number25Player nameBen Slimane
Average rating
4.69
Line-upsMatch StatsLive TextLine-upsLiverpool
Formation 4-3-3
62Kelleher
84Bradley5Konaté4van Dijk2Gomez
8Szoboszlai10Mac Allister38Gravenberch
11Salah9Núñez7Díaz
62Kelleher84Bradley5Konaté4van Dijk2GomezSubstituted forGakpoat 73'minutes8SzoboszlaiSubstituted forJonesat 73'minutes10Mac Allister38GravenberchSubstituted forRobertsonat 60'minutes11SalahSubstituted forElliottat 60'minutes9Núñez7DíazSubstitutes13Adrián17Jones18Gakpo19Elliott21Tsimikas26Robertson42Clark76Danns78QuansahSheff Utd
Formation 3-5-2
13Grbic
30Holgate15Ahmedhodzic19Robinson
20Bogle21de Souza Costa24Arblaster8Hamer5Trusty
28McAtee11Brereton
13Grbic30HolgateSubstituted forBen Slimaneat 87'minutes15Ahmedhodzic19Robinson20Bogle21de Souza Costa24Arblaster8HamerSubstituted forOsbornat 62'minutes5TrustySubstituted forBrooksat 78'minutes28McAteeSubstituted forMcBurnieat 63'minutesBooked at 87mins11BreretonSubstituted forOsulaat 63'minutesSubstitutes9McBurnie16Norwood18Foderingham23Osborn25Ben Slimane27Larouci29Curtis32Osula35BrooksReferee:Stuart AttwellAttendance:60,055Live TextPost update
Match ends, Liverpool 3, Sheffield United 1.
Full Time
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