The two bitter rivals were meeting on the European stage of the first time ever, and goals from Daniel Sturridge and Roberto Firmino gave the home side a significant advantage at the halfway stage of the tie.
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It took just 15 seconds for the first chance to arrive as United came forward early down the left, but Marcus Rashford did not expect Alberto Moreno to miss Memphis Depay's cross and could not react in time to make anything of the opportunity.
Despite that quick start, it took until the 20th minute for the first shot of the game, and it was enough to give Liverpool the lead as Sturridge converted from the spot.
Depay was the man penalised for a tug on Nathaniel Clyne right on the edge of the penalty area and, after referee Carlos Velasco had pointed to the spot, Sturridge placed his finish past David de Gea from 12 yards.
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The United keeper did get a hand to the spot kick without being able to keep it out, but the Spaniard kept his side in the tie over the remainder of the first half with three saves to thwart golden opportunities for Liverpool.
The first came from Philippe Coutinho as the Brazilian latched on to Sturridge's dragged effort at the back post, but with the goal gaping he tried to nudge the ball over the line with the outside of his boot, scuffing it and allowing De Gea to get back and claw it away off the line.
A slip from Chris Smalling when trying to clear the ball then saw Sturridge presented with a clear chance on the left side of the area, but De Gea was quick off his line to make himself big and save the Liverpool's striker's powerful effort.
The Spaniard completed his trio of stops with three minutes to go until the break when Firmino's cross was flicked towards goal from close range by Adam Lallana, drawing a reaction stop from the keeper.
United finally had their first shot of the contest in the 51st minute when Morgan Schneiderlin brought a routine stop from Simon Mignolet, but it wasn't long before De Gea was once again forced to come to the visitors' rescue, this time tipping Coutinho's powerful effort over the top.
Another save from a swerving Clyne effort followed, before Jordan Henderson steered a shot of his own narrowly past the far top corner from the edge of the box.
The hosts did finally get a second goal with just under 20 minutes remaining, however, when Michael Carrick's error gifted Lallana the ball inside the area, and he kept his composure to find Firmino, who swept home the finish from close range.
United's most threatening moment of the entire match eventually arrived in the 85th minute when Marouane Fellaini climbed above Dejan Lovren inside the area, but he nodded his effort a yard past the post.
There was to be no reply from the visitors, though, as Liverpool ended a four-game losing streak against their biggest rivals to put one foot in the quarter-finals.