Reduced to 10 men for the closing stages after Fabio was sent off, Swansea's battering ram striker Wilfried Bony climbed high to head home a powerful 90th-minute winner and leave United boss David Moyes to face questions about his management.
Forest's Jamie Paterson netted a hat-trick in the humiliating rout, but the spotlight was on Allardyce who was accused of waving the white flag by making nine changes to his starting line-up with an eye on the League Cup semi-final at Manchester City on Wednesday.
Chelsea showed their fellow Londoners how to negotiate a potentially tricky Cup tie in the east Midlands, winning 2-0 at second tier Derby County.
A chorus of boos rang around the ground as Swansea became the fourth team to walk away victorious in United's last six home encounters in all competitions, a run of results that would have been almost unthinkable under Ferguson.
"We were a bit unlucky to lose but probably did not do enough to win in the end," Moyes told BT Sport.
"If you lose games at any club it's a big deal but at Manchester United it's even bigger. We have got a big game on Tuesday (League Cup semi-final first leg against Sunderland), we must get the players ready for that."
It was the first time Swansea had tasted victory at United's home ground, but it was far from a smash-and-grab raid.
Having lost their fourth home Premier League match last week against Tottenham Hotspur, the Old Trafford fans were again fearing the worst when Swansea went ahead with a lobbed finish from Wayne Routledge after 12 minutes.
Javier Hernandez restored parity with a close-range finish four minutes later, but that did not prove to be a platform on which the hosts could build.
The second half was not easy on the eye and was only sparked into life when substitute Fabio was sent off for a lunging, studs-up challenge on Jose Canas.
Late goals were United's speciality under Ferguson, but having only scored more than one goal once in their last six at home, Swansea were the more likely victors.
So it proved when Bony rose unmarked at the near post to head Routledge's cross into the top corner.
West Ham, 19th in the top flight and facing a relegation battle, were undone by a Forest side who scored early in the first half with a cheeky Djamel Abdoun penalty before Paterson netted three in the second half and Andy Reid added a fifth.
"Who knows, that's their decision, not mine," a beleaguered Allardyce told reporters when asked whether the defeat would have repercussions for him.
"I am not saying it won't happen, unless I start getting results, I have got to get results I know that."
Forest's local rivals Derby had little joy against Chelsea, who named a strong side with enough class to dominate their lower league opponents.
John Obi Mikel put them ahead after 66 minutes, the Nigeria international getting on the end of a Willian set-piece to head in his fourth goal for Chelsea in almost 300 appearances.
Oscar added a second from the edge of the area to make the game safe and set up a fourth-round tie at home to Premier League Stoke City.
Liverpool suffered a painful defeat by Oldham in the fourth round last season, a reverse manager Brendan Rodgers described as the lowest point in his Liverpool career.
The potential for another embarrassing defeat was reduced nine minutes into the second half when Aspas steered home Raheem Sterling's cross and James Tarkowski inadvertently redirected a Sterling shot into his own net to double the lead.
Sunderland were briefly made to sweat when Adam Johnson's opener was cancelled out by Carlisle's Matty Robson just before halftime.
The second half was just five minutes old, however, when a comical own goal restored the Premier League side's lead, with Sean O'Hanlon sliding in to divert Johnson's cross into the net.
El Hadji Ba made the game safe on 90 minutes when he side-footed home from close range.
The lower league clash between Port Vale and Plymouth Argyle ended in a 2-2 draw and will go to a replay.