Steven Gerrard was sent off as Ryan Giggs's second-minute penalty proved enough to send Manchester United into the fourth round of the FA Cup, ruining Kenny Dalglish's emotional return as Liverpool manager.
The tie was barely 30 seconds old when Daniel Agger was adjudged to have fouled Dimitar Berbatov in the box and Giggs converted the spot-kick with ease.
Liverpool responded well, but their hopes of a comeback were hit hard when Gerrard was shown the red card for diving in rashly on Michael Carrick just after the half-hour mark.
Despite a spirited effort that offered encouragement to the 9,000 visiting fans, only a succession of fine saves by Pepe Reina kept the game alive until the end.
Roy Hodgson's departure on Saturday had offered Dalglish the opportunity to renew his old rivalry with Sir Alex Ferguson and added extra spice to an already enticing encounter but, 20 years on from his last spell in charge of the Merseysiders, he could not inspire an instant improvement in their fortunes in his first game back in the dug-out.
Dalglish will have easier games in which to make his mark than this one between now and the end of the season.
However, his second stint as Liverpool manager got off to the worst possible start when Berbatov ambled into the box and went to ground near the right byeline after Agger clipped his ankle.
Referee Howard Webb thought long and hard before finally deciding it was a penalty and the veteran winger Giggs made no mistake, stepping up to slam the ball into the corner of the net.
Liverpool initially responded well, showing plenty of the heart they were purported to lack under Hodgson and fighting well to establish a platform in midfield through Lucas and Raul Meireles.
Although they were hardly piling on the pressure, the visitors tested the United defence too, with Rio Ferdinand having to show nifty footwork to clear from the edge of the six-yard box as Fernando Torres looked to pounce on Martin Kelly's low cross.
Maxi Rodriguez and Gerrard also both brought decent saves from Tomasz Kuszczak and - with United offering little at the other end - the game appeared to still be in the balance until Gerrard's moment of madness.
There was absolutely no need for the Reds skipper to dive in two-footed with studs showing, even if he did get far more of the ball than the man. Again Webb gave his decision some thought but, when it came, the red card he brandished was no surprise.
United, who had barely got out of second gear, were now in control of the game and almost extended their lead on the stroke of half-time when Jonny Evans ran in to meet a floated Giggs corner and power a header against the near post.
The home side continued to come forward with purpose at the start of the second half as Javier Hernandez flashed a header wide before Berbatov blasted over when Reina fumbled a Giggs cross.
Liverpool's 10 men continued to battle on and went close when Kuszczak had to dive acrobatically to keep out Fabio Aurelio's free-kick, which was heading for the top corner.
But, apart from an Agger header that was easily saved, Dalglish's side rarely threatened an equaliser while United were enjoying plenty of chances at the other end.
Reina made three excellent saves in quick succession from Rafael, Berbatov and Patrice Evra, with Giggs having a decent penalty shout turned down in between, while substitute Michael Owen - a former Anfield star - also spurned a couple of late opportunities as United ended up convincing winners.
Defeat means that only the Europa League offers Dalglish the chance to mark his second spell in charge at Anfield with a trophy but, to the Reds fans who chanted his name until long after the final whistle at Old Trafford, his presence means they can hope for a brighter future.