Manchester: Trailing 2-1 from the first leg, United levelled the tie through Jonny Evans's 37th-minute header, only for Phil Bardsley to net a dramatic equaliser late in extra time with a low shot that squirmed past David de Gea.
United promptly went straight up to the other end and equalised through Javier Hernandez, for 3-3 on aggregate and sending the tie to penalties, but it was to prove a false dawn.
Sunderland players after their win. Pic/ AFP
An appalling shoot-out saw only three of the 10 penalty-takers score, but United full-back Rafael da Silva proved the fall guy, with Vito Mannone saving his spot-kick to give Sunderland a 2-1 win.
Second from bottom in the Premier League, Gus Poyet's side can now look forward to a clash with Manchester City in the final on March 2, in what will be their first major final since 1992.
For David Moyes, meanwhile, the misery continues, with a first final as United manager snatched from his grasp only days after defeat at Chelsea saw his side fall 14 points off the pace in the league.
United's fans arrived at Old Trafford to reports that their side have made a club-record bid for Chelsea midfielder Juan Mata, and the continued absence of Wayne Rooney and Robin van Persie due to injury underlined the need for reinforcements.
There was an early chance for Hernandez, whose downward header from Adnan Januzaj's free-kick was parried by visiting goalkeeper Mannone, but the hosts initially struggled to make inroads.
Nine thousand Sunderland fans travelled from the northeast and the hordes of away supporters almost had a memorable goal to cheer when Fabio Borini chested down a raking pass from Adam Johnson and drilled a half-volley inches over the bar from 25 yards.
Moyes reacted by bringing Shinji Kagawa into the centre and moments later the Japan midfielder created the chance that led to the opening goal.
His cross was volleyed against the post by Darren Fletcher, with Mannone blocking Hernandez's follow-up, and from the ensuing corner, Januzaj's cross was flicked goalwards by Danny Welbeck and headed in by Evans.
As Sunderland grew bolder in the second half, Moyes introduced Antonio Valencia in place of Kagawa and the Ecuadorian quickly created a chance for Januzaj, only for the teenager to slice wide.
It was Sunderland who continued to force the issue, though. Only a flying block from Alexander Buttner thwarted Johnson, while Marcos Alonso almost found the net with an audacious volley that zipped just past the post.
Chris Smalling, meanwhile, was fortunate to avoid punishment when he dragged Sunderland striker Steven Fletcher down after misjudging a bouncing ball.
An injury-time free-kick gave Januzaj a chance to snatch a winner, but his chipped effort was comfortably saved by Mannone, sending the game to extra time.
United sustained a blow when Michael Carrick had to go off after twisting his ankle, with Phil Jones coming on, and they wasted a chance to increase their lead when Hernandez shot wide after Januzaj put him clean through.
The game appeared up for United when De Gea somehow allowed Bardsley's low daisy-cutter to squirm through his grasp and into the net in the 119th minute.
United were stunned. Just seconds later, Januzaj crossed for Hernandez to slam home an even more dramatic goal, but the night would belong to Sunderland.