United competed until the 59th minute when Mike Williamson fouled Luis Suarez for a penalty that Dirk Kuyt converted.
Elsewhere, Liverpool cruised to a 3-0 win over Sunderland thanks to two goals from Luis Suarez.
Its wingers have been inconsistent, and its other striker, Luis Suarez, hasn't shown much chemistry with Carroll.
Luis Suarez also scored a hat-trick on Saturday as Liverpool trounced Wigan 4-0 at the DW Stadium.
Liverpool footballer Luis Suarez has courted controversy yet again, this time by biting an opponent on the arm.
Liverpool striker Luis Suarez was twice denied a goal by the woodwork on his first start for the club.
But two goals before halftime gave Liverpool hope as Luis Suarez and Joe Allen made it 3-2 on aggregate.
Luis Suarez set him up for his second after the break and he rounded off his triple in stoppage time.
Chelsea's John Terry and Liverpool's Luis Suarez were fined and received lengthy bans for two separate offenses which occurred in 2011.
"Adidas takes this type of incident very seriously and does not condone Luis Suarez's behavior, " it said in a statement Monday.
Striker Luis Suarez was handed an eight-game ban by the English Football Association for racially abusing Manchester United's French defender Patrice Evra.
"I'm sure people will want to talk about Luis Suarez, but he accepted his ban, served his punishment and returned, " he said.
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In December, Liverpool striker Luis Suarez was photographed giving an American-style middle finger to Fulham fans after his club's 1-0 loss there.
Uruguay international Luis Suarez was at the center of the action, his early shot deflecting off Leighton Baines to produce the opening goal.
Uruguay talisman Luis Suarez dragged his side back into the tie, with a key role in both Gaston Ramirez's equalizer and Nicola Lodeiro's winner.
Kuyt's hat-trick was the first for Liverpool against United since Peter Beardsley's in September 1990 as Luis Suarez had a hand in all three goals.
The Reds, buoyed by scoring, came close to adding another when Luis Suarez's low half volley had to be palmed away by the Sunderland keeper.
Liverpool's Luis Suarez damaged Chelsea's hopes of a top-four finish Sunday afternoon with an equalizer in the seventh minute of stoppage time to secure a 2-2 draw.
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Luis Suarez refuses to shake the hand of Patrice Evra and sets of a furor at the start of the Premier League match between Manchester United and Liverpool.
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My friend and peer at work, Luis Suarez, would probably also like to point out: consider reducing how you interact with email, and favor these alternatives instead.
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Luis Suarez, who was serving the first of a 10-game ban for biting Chelsea's Branislav Ivanovic, was barely missed as the Reds ran riot at St James Park.
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Striker Luis Suarez has been charged with violent conduct by the Football Association and is facing a ban for biting Chelsea defender Branislav Ivanovic on the arm.
Liverpool looked a shadow of the side which beat Manchester United 3-1 last Sunday as they missed captain Steven Gerrard through injury, while striker Luis Suarez was ineligible.
Luis Suarez, who has now bitten people in two leagues, crossed a line of decorum and crassness that sees punching and kicking on the other, more proper, side of it.
Liverpool's Uruguay international striker Luis Suarez has been accused of giving "unreliable" and "inconsistent" evidence to the disciplinary panel which banned him for eight-games for alleged racial abuse of Manchester United's Patrice Evra.
The Atletico Madrid striker was alive with craft and invention, but he wasted a decent opening with 20 minutes left, shooting wide from 16 yards after a decent lay-off from Luis Suarez.
The game finished 2-2 with Luis Suarez equalizing with the last attacking touch of the game and in the process Liverpool whisked two very valuable points out from under the nose of Chelsea.
Over the last year English football has had to grapple with the fall out of racism cases involving Liverpool striker Luis Suarez and Chelsea defender John Terry, with English FA punishing both players.
Saturday's FA Cup match was the first time the two teams had met since a racist incident between Manchester United's French defender Patrice Evra and Liverpool striker Luis Suarez during their league encounter last year.
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