Here's a selection of tweets reacting to the result, commenting on United's form and addressing Moyes' position at United.
Moyes says #mufc were the better team vs Swansea but didn’t create enough chances, adding: “Our results at home haven’t been good enough.”— Manchester United (@ManUtd) January 5, 2014
At 30th Sept 2013 @ManUtd had cash of £83.6m in the bank and will generate another £80m this season even if finishing outside CL places— Andy Green (@andersred) January 5, 2014
What did Manchester United fans expect? The club appointed a manager with a track record of winning nothing.— Sam Pilger (@sampilger) January 5, 2014
If Man Utd sack Moyes they surely must also sack the £2m-a-year adviser who insisted he become manager.— Sachin Nakrani (@SachinNakrani) January 5, 2014
For the first time in 24 seasons, Manchester United have lost five home matches before the month February.— Infostrada Sports (@InfostradaLive) January 5, 2014
2 - Man Utd have been eliminated at this stage of the #FACup for just the second time in 29 years (also 2010 v Leeds United). Surprise.— OptaJoe (@OptaJoe) January 5, 2014
FT: Man Utd 1 Swansea 2. #MUFC out of the cup and deservedly so. Abject. The boos nothing like as loud as you'd expect. Most in shock— Oliver Kay (@OliverKayTimes) January 5, 2014
Another damaging home defeat for Man Utd and manager David Moyes as Swansea City win 2-1 in the FA Cup third round.— Phil McNulty (@philmcnulty) January 5, 2014
The Swans have dumped @ManUtd out of the FA Cup at Old Trafford. Fully deserved for Laudrup's men. Fantastic.— Swansea City FC (@SwansOfficial) January 5, 2014
Even if Manchester United were to win all of their remaining PL games they would still finish on fewer points than last season #MUFC— EPL Stat Man (@EPLStatman) January 5, 2014
My fellow United fans - stop moaning - we've not really looked any good for over 5 years despite the titles #fact— terry christian (@terrychristian) January 5, 2014
United backing Moyes with time alone is not enough. That's unfair. He needs to be backed with both time AND money. That's fair.— Nooruddean (@BeardedGenius) January 5, 2014
Up next for United is a trip to face Sunderland in the Premier League.
At least it's not at Old Trafford.