Goals from Daniel Sturridge and Roberto Firmino put Liverpool firmly in command of this Europa League tie with United indebted to a string of fine saves from David de Gea.
The Reds face an uphill struggle in next week's second leg at Old Trafford but have De Gea to thank for keeping us in with any semblance of a chance after a virtuoso performance.
As expected, the contest started at a frenetic pace in front of a raucous atmosphere and United immediately set about silencing the fervent Anfield crowd.
Marcus Rashford almost converted a Memphis cross on the stretch, but the first real opening of the match led to the breakthrough on 19 minutes.
Nathaniel Clyne surged forward from right-back and went down under a challenge from Memphis and, despite De Gea getting a hand to it, Sturridge slotted home the spot-kick with aplomb.
That goal sparked Liverpool into life and they looked certain to go 2-0 ahead just minutes later, only for De Gea to claw the ball away from point-blank range to keep out Philippe Coutinho.
The Spaniard was called into action again shortly after the half-hour mark to deny the England international after Chris Smalling's slip had let him in, while Memphis and Marcos Rojo both blasted off-target from distance at the other end.
Liverpool were not finished before the break, though, as Emre Can crossed for Adam Lallana, but De Gea once again came to United's rescue to excellently repel the volley.
Simon Mignolet was finally tested by Morgan Schneiderlin shortly after the break, but the hosts remained on the front foot and almost doubled their lead twice in quick succession on the hour mark.
De Gea tipped over from Coutinho when the Brazilian's dipping drive seemed destined for the top corner, and then once more reacted superbly to keep out Clyne when he palmed away the full back's fizzed effort.
There was nothing he could do to prevent the second goal, which had been a long time coming and duly arrived on 73 minutes.
Substitute Michael Carrick's attempted clearance fell short and the ball dropped to Lallana whom cut it back for Firmino to roll home from six yards.
Despite the two goal deficit, United knew that an away goal would change the complexion of the tie, and the recalled Marouane Fellaini went closest to it with a header from substitute Bastian Schweinsteiger's cross but that was the best we could muster.
Overall team performance: No rating
United Faithful Man of the Match: David de Gea. Magnificent and the only reason we're still in this.