This was Liverpool's first home match since an independent report cleared their fans of any responsibility for the 1989 Hillsborough disaster where 96 of the Merseysiders' supporters were crushed to death during an FA Cup semi-final.
Liverpool played more than half the match a man down after Jonjo Shelvey was sent off in the 39th minute for a two-footed challenge on Jonny Evans.
But they still took a 46th minute lead when captain Steven Gerrard – whose 10-year-old cousin, Jon-Paul Gilhooley, was the youngest fan to die at Hillsborough – volleyed home from near the penalty spot.
However, United only had to wait five minutes for an equaliser when Rafael curled in a superb shot from the right-hand side of the Liverpool area.
And with 15 minutes to go United – who had missed their last three penalties – were awarded a spot-kick when Liverpool defender Glen Johnson brought down Antonio Valencia.
There was a delay of several minutes as Liverpool defender Daniel Agger received treatment for an injury before Dutch striker Robin van Persie beat goalkeeper Pepe Reina.
Victory saw United move to within a point of leaders Chelsea while defeat left Liverpool languishing in the bottom three.
Before kick-off, a potential flashpoint was avoided when Liverpool striker Luis Suarez and United defender Patrice Evra shook hands.
Suarez, banned for eight games for racially abusing Senegalese-born Frenchman Evra in the equivalent fixture last season, declined to shake the United man's hand before the clubs' last meeting in February.
The Hillsborough victims were remembered in a series of pre-match ceremonies.
United legend Bobby Charlton presented former Liverpool striker Ian Rush with a bouquet of red roses, and United captain Ryan Giggs released 96 red balloons, one for each of the dead.
Tributes to the Hillsborough dead climaxed with the words "The Truth", "Justice" and "96" being spelled out by fans holding red and white cards.
Sunday's other Premier League matches see champions Manchester City at home to high-flyers Arsenal while Chris Hughton takes Norwich to St James' Park for the first time since being sacked as Newcastle manager and Tottenham Hotspur face Queens Park Rangers in a London derby at White Hart Lane. – Sapa-AFP