Three points were required from the Merseyside giants if they were to return to the top of the table at Anfield, having been bumped down to second spot by virtue of Arsenal's win over Luton Town on Wednesday.
Liverpool struggled to find their rhythm in the early stages, but they nevertheless went into the break a goal to the good in bizarre circumstances, as Darwin Nunez charged down Ivo Grbic's clearance and saw the ball roll into the empty goal off his backside.
However, Klopp's men were still a shadow of their scintillating best, and Chris Wilder's basement boys pegged them back 13 minutes into the second half, as Conor Bradley diverted Gustavo Hamer's header into his own net.
The Blades' reprieve was short-lived, though, as the ball broke kindly for Mac Allister on the edge of the box midway through the second period, and the Argentine unleashed a rocket into the top corner in front of the Kop.
Klopp on "absolutely insane" Mac Allister display
The midfielder's astounding effort evoked memories of his equally wonderful strike against Fulham earlier in the campaign, and after Mac Allister had reignited the Reds' fire, Cody Gakpo netted the insurance goal with seconds of normal time remaining.
In between his screamer and Gakpo's header, Mac Allister also crashed a stunning free kick off the bar, and Klopp credited the ex-Brighton & Hove Albion man with scoring the best two goals in the current Premier League campaign.
"He is a really good player, in the first half he was playing six but we knew we could move him slightly higher," Klopp told BBC Sport, having been forced to move Mac Allister into a deeper role owing to Wataru Endo's absence through injury.
"Then he scores this kind of goal, the two biggest screamers of the season have come from him. Then the free-kick directly after - absolutely insane, what a player."
While Mac Allister's thunderbolt got the Anfield crowd off their feet, the Merseyside atmosphere was otherwise subdued as Liverpool struggled to break down their well-drilled opponents, who are scrapping for their lives at the foot of the division.
Klopp makes "not great" admission as Liverpool survive scare
Klopp admitted that his side's football left a lot to be desired in certain moments, but he credited his side's slick passing sequences for their improved display in the closing stages, adding: "We had to dig very deep, but that is normal. Our attitude was good, the football was not great in a lot of moments.
"We saw how quick passes can change the game after 60 minutes, Robertson down the line, bam bam bam. Then Macca's goal [Alexis Mac Allister] was the complete game changer, and it's a wonderful goal for 3-1 as well. The first goal, Darwin does it by himself.
"They could have scored after 20 seconds, Caoimhin [Kelleher] makes a great save. Sheffield United played how they wanted for 90 minutes. It is difficult, but we could do better, and we did to get the win.
"I am happy we play on Sunday, we cannot change that. I don't see the problem. The game we played tonight was nothing to do with yesterday, and Sunday will be nothing to do with the other results."
Now two points clear of Arsenal, Liverpool return to action away to Manchester United on Sunday, by which time the Gunners and Manchester City would have already taken to the field.