While Chelsea progressed with a 1-0 win at Queens Park Rangers thanks to Juan Mata's second-half penalty, Netherlands forward Dirk Kuyt was the match-winner for Liverpool in the 88th minute at Anfield.
With United having dumped out neighbour Manchester City — the Premier League leaders and defending FA Cup champions — in the fourth round, the world's oldest club knockout competition looks wide open this year.
Second-tier Brighton beat Premier League Newcastle 1-0 at the Amex Stadium in another Cup match that Magpies defender Mike Williamson will want to forget.
Williamson deflected in Will Buckley's close-range effort for the only goal 14 minutes from time. The defender also scored an own goal last season when Newcastle lost to then League Two side Stevenage in the third round of the competition.
Bolton beat Swansea 2-1 and Norwich won by the same scoreline at West Bromwich Albion in the other all-Premier League matchups, while Stoke — which lost the 2011 final to Man City — also progressed with a 2-0 win at Derby.
Tottenham and Everton went through on Friday, while Arsenal hosts Aston Villa on Sunday.
On a day without any major shocks, fourth-tier team Crawley provided the biggest upset by beating Hull, which is fifth in the second tier, 1-0. There are 45 places between the teams in English football's pyramid.
Liverpool-United was undoubtedly the biggest game of the day and the rivals' first meeting since the unsavoury race row between Luis Suarez and Patrice Evra erupted in a Premier League match between the teams in October.
Evra, United's captain on Saturday, was booed throughout while Suarez was unavailable as he served the seventh of his eight-game ban for repeatedly racially abusing the France defender.
The match passed without trouble, however, with United manager Alex Ferguson saying: "The players showed great respect to each other — there wasn't a bad tackle in the game."
Denmark centre back Daniel Agger's opener for Liverpool in the 21st minute was cancelled out by United's Park Ji-sung six minutes before the break in a first half edged by the visitors, despite being without a raft of key players including the injured Wayne Rooney.
Kuyt settled the match when he ran to a flick-on by Andy Carroll and beat United goalkeeper David de Gea at his near post.
It was a perfect end to a week that saw Liverpool also reach the final of the League Cup, against Cardiff on Feb. 27, by completing a two-leg victory over City on Wednesday.
"It's a fantastic credit to the players and how much pride they take in the club that they've got themselves back on track," said Liverpool manager Kenny Dalglish, who criticized his team for a sloppy 3-1 league defeat to Bolton last weekend. "They've come out on the right side of two huge cup games."
Victory means Liverpool is unbeaten in its last three matches against United.
Chelsea's trip to QPR was also haunted by the backdrop of racism, with the home fans taunting Blues captain John Terry over allegations he racially abused QPR defender Anton Ferdinand when the teams last met in October.
Ferdinand, the younger brother of Terry's England centre-back partner Rio, was spared having to decide whether to give Terry a pre-match handshake after the Football Association allowed the sides to forego the traditional ritual before the match.
Terry, who faces a criminal charge over the racism claim and will appear in court Wednesday, seemed unaffected in a game that Mata clinched from the spot after Clint Hill was adjudged to have brought down Daniel Sturridge.
Chris Eagles scored a second-half winner as Bolton came from behind to see off Swansea, while Simeon Jackson's 85th-minute goal secured Norwich's victory over West Brom in a repeat of the result between the teams a fortnight ago.
Goals from Cameron Jerome and Robert Huth bookended Stoke's win at Derby.