David De Gea fumbled Phil Bardsley's effort before Javier Hernandez forced penalties, but the hosts produced some terrible spot kicks and the visitors will meet Manchester City.
The rumored United target saved Rafael's crucial penalty in a dismal shootout which saw just three players find the back of the net after the hosts had won an incredible second leg 2-1.
Jonny Evans put the hosts ahead in the 37th minute with a header from close range, but neither side could find the quality to breach either defense thereafter.
With away goals not coming into force until the end of extra time in the Capital One Cup, United was required to hang on to its narrow lead for an additional 30 minutes, but looked to be out when David de Gea could only shovel Phil Bardsley's strike into the net.
|VIEW FROM OLD TRAFFORD
|By Tom Maston
With little over two minutes of extra time left at Old Trafford, David Moyes would have been forgiven for looking forward to his first major cup final at Wembley as Manchester United manager.
But after a horrible error from his own goalkeeper and heroics from Vito Mannone, he must now reflect on a huge missed opportunity. His side really should have sewn it up before then, but United's need for someone of Juan Mata’s ilk was shown up for all to see.
For Sunderland and its 9,000 traveling fans, a place at Wembley was great reward for two lung-busting displays across both legs, and Gus Poyet's men can rightly feel like they deserve their place alongside Manchester City in early March.
In a frantic last two minutes, Javier Hernandez then sent the tie to penalties from close range, but with both teams short of quality from spot, Mannone made the vital saves to book Sunderland a spot in its first League Cup final since 1985.
Moyes made five changes to the United side that lost the tie’s opener 2-1, with Darren Fletcher and Hernandez among the players to come in.
Chicharito was denied early on as his headed effort from an Adnan Januzaj free-kick was palmed away by Mannone.
With chances at a premium, Fabio Borini lashed a speculative half-volley over, while Darren Fletcher hit the post at the other end, before Danny Welbeck’s follow-up was cleared.
But Moyes’ men were ahead moments later, as two former Sunderland loanees combined to level the tie. Welbeck flicked the ball on from a corner, teeing up Evans to head home from inside the six-yard box.
Rafael went agonizingly close to giving United an aggregate lead four minutes into the second half, hooking a left-footed strike just over the crossbar from the edge of the penalty area.
Januzaj was next to try his luck from distance, drilling just wide of the right-hand upright in the 55th minute, but both teams struggled to break down the opposition defense as the half progressed, with extra time looking an increasingly likely prospect.
Sunderland had cause to feel harshly done by in the closing stages of normal time when Chris Smalling escaped punishment for a tug on Steven Fletcher, before Marcos Alonso blazed narrowly wide of the right-hand post from a tight angle.
Bardsley looked to have sent Sunderland into the final when De Gea spilled his 20-yard effort into the net in the 119th minute, but Hernandez and Januzaj exchanged passes well before the former smashed home from close range to force penalties.
Alonso, Ki Sung-Yueng and Darren Fletcher netted in an otherwise dismal showing among the first eight spot kicks.
Adam Johnson appeared to have wasted Sunderland's chance to win as De Gea saved his effort from 12 yards, but Rafael saw the following spot-kick saved as Mannone earned victory for Sunderland, which now meets Manchester City at Wembley on March 2.