Steven Gerrard's volleyed opener - after Jonjo Shelvey had been sent off - was cancelled out by Rafael's curled equaliser before the Dutchman scored the winning penalty
Jonjo Shelvey's red card six minutes before half-time threatened to derail the attacking dominance that Brendan Rodgers' side enjoyed for the first 40 minutes, but Steven Gerrard's volley propelled them into the lead in the first minute of the second-half.
The visitors responded with an immediate equaliser - right-back Rafael curling a corker into the top corner from the right side of the penalty area.
The 10 men of Liverpool continued to battle but, with 10 minutes left, Robin van Persie ended United's recent run of poor penalty-kicks by converting from the spot after Antonio Valencia was fouled to secure victory for Sir Alex Ferguson's side and leave the hosts stranded in the relegation zone.
The theme before the game was unity. The two managers, Ferguson and Brendan Rodgers, conducted a joint interview and Patrice Evra and Luis Suarez shook hands despite their embittered history. On the pitch and in the stands, however, the passionate rivalry was as fierce as ever.
Buoyed by a rousing rendition of 'You'll Never Walk Alone' in memory of the 96 Liverpool fans that lost their lives at Hillsborough 23 years ago - 11 days after the Hillsborough Independent Panel's report had definitively absolved Reds' supporters of all blame for the tragedy - Liverpool were by far the stronger team from kick-off.
Suarez forced Anders Lindegaard into an early save from close range before an unmarked Steven Gerrard rippled the site-netting with a low shot from a smartly-worked corner.
The Reds' attacking continued as United's passing accuracy plummeted and Suarez was at the centre of most moves, giving centre-backs Rio Ferdinand and Jonny Evans a torrid time.
The home side suffered a major set-back on the 39-minute mark, though, when Shelvey was sent off for a lunging, studs-up challenge on Evans. The young midfielder proceeded to direct some choice words in the direction of Ferguson as he left the field.
If any Reds supporters were worried that their team would lessen their grip on the game, however, those fears were temporarily dispelled when Gerrard chested the ball down 15 yards from goal and rifled a volley home, sparking scenes of jubilation around Anfield.
Unfortunately for Rodgers' team, Rafael demonstrated why United should never be underestimated regardless of their performance level by curling sublimely into the top left-hand corner to level the scoring five minutes after Gerrard had struck.
Paul Scholes - a half-time replacement for the anonymous Nani - helped bring stability and passing sensibility to a United midfield which lacked mobility in the first-half.
Suarez, an energetic force throughout, forced a super save to Lindegaard's right with a low effort, though, and substitute Suso tested the Danish goalkeeper again from just inside the box.
But the home side’s hopes were dashed when Glen Johnson clattered into the back of Antonio Valencia as United counter-attacked and gave away a penalty in the visitors' favour.
Having missed from the spot at Southampton earlier this season, Van Persie made no mistake in driving what proved to be the winning goal past goalkeeper Pepe Reina, who got a hand to the ball but could not tip it wide.
At the other end, Martin Kelly was gifted a gilt-edged chance to grab a dramatic equaliser but his header inside the box from Johnson's cross flew wide as the Reds were left languishing in the bottom three.