The penalty and 32nd-minute dismissal of Liverpool's captain enabled United to cruise through to a fourth-round away game with League Two Southampton and spoil the comeback of Dalglish, who replaced Roy Hodgson on Saturday.
Holders Chelsea progressed with a 7-0 thrashing of Ipswich Town at Stamford Bridge and will visit Everton, while Tottenham Hotspur beat League One Charlton Athletic 3-0 and Manchester City were held 2-2 at Leicester City.
The match at Old Trafford was already the stand-out tie of the third round but the return of Dalglish added extra spice.
However, all the hopes pinned by Liverpool fans on "King Kenny" lasted about 30 seconds as recalled centre-back Daniel Agger tripped Dimitar Berbatov and Giggs rammed home the penalty.
The 9,000 travelling fans continued to give noisy support and helped produce a terrific atmosphere but they were silenced again when Gerrard was shown a straight red card for a studs-up challenge on fellow England midfielder Michael Carrick.
Johnny Evans smacked a header against a post just before half-time, Javier Hernandez nodded inches wide just after and both men went close during a mad scramble on the hour as United searched for the second.
The 11-times winners could not find it but Liverpool were unable to muster any threat.
"I think the most important thing for us was that we got a great deal of effort and commitment," Dalglish told reporters.
"They had enthusiasm and endeavour for the cause, which was helped and matched by the supporters who came along here. If we get everyone pulling in the one direction we are going to have a good chance of improving things.
"The two decisions are important factors in the game, added Dalglish, who said he disagreed with both.
"But it's fantastic to be back."
United boss Sir Alex Ferguson told MUTV: "It was a sensible performance. We didn't do anything stupid and against the 10 men we worked the ball about and came through easily enough."
Carlo Ancelotti, another manager under pressure, will have been delighted to have seen his struggling Chelsea side notch their biggest Cup win for more than a century against managerless Ipswich, who sacked Roy Keane last week.
Two goals in two minutes just after the half-hour by Salomon Kalou and Daniel Sturridge, with a neat backheel, got things moving and a Carlos Edwards own goal made it 3-0 at half-time.
Nicolas Anelka, a second for Sturridge then another two in two minutes for Frank Lampard completed the thrashing.
Spurs went ahead against Charlton four minutes into the second half through 19-year-old debutant Andros Townsend.
Jermain Defoe then cut through the visiting defence twice in three minutes to earn a fourth-round trip to Fulham.
Leicester shocked City by taking a first-minute lead through debutant centre back Sol Bamba. James Milner levelled after 23 minutes and C
