City went into the contest with a 3-1 advantage from the first leg, but Nemanja Matic got United back into the tie with a superbly-taken volley before the break.
However, despite failing to score in front of their own supporters, City were able to see out the rest of the contest, helped by the dismissal of Matic for a second yellow card.
Pep Guardiola's side will now square off against Aston Villa in the final on March 1 as they look to win this competition for the third time in succession.
Moments later, Riyad Mahrez cut inside onto his left foot before seeing his deflected effort well turned away by De Gea, who was needed to keep his team in the tie with his backline struggling to cope with City's forward players.
While City stopped pushing relentlessly after the quarter-hour mark, they were still creating opportunities, with De Gea again being required to claw away Aguero's rising shot which was heading into the roof of the net.
United were offering very little in the way of a response, recording no shots on goal during the opening 30 minutes, but that soon changed as Matic put the visitors in front.
A free kick from the left was only half cleared into the path of the midfielder, who sent a volley into the bottom corner of Claudio Bravo's net via the inside of the post.
The home crowd were stunned and growing increasing tense, and that soon turned to disappointment when Raheem Sterling had an equaliser disallowed due to the winger being a yard offside when latching on to a Kevin De Bruyne cross.
Aware that they had their hosts on the rack, United pushed forward after the restart, and Harry Maguire should have done much better with a header at the back post after being found by an inswinging Fred free kick.
However, after successfully withstanding more pressure from United, City began to find space of their own at the other end, with Sterling failing to convert from 12 yards out after biding his time inside the area.
While the momentum had swung in City's favour, they were struggling to create clear-cut opportunities, leaving Kyle Walker to try his luck from 30 yards with a strike which De Gea punched clear.
Despite somehow managing to keep City out until this point, United's challenge was made even tougher with 14 minutes remaining as Matic was handed a second yellow card for a cynical foul on Ilkay Gundogan, deemed necessary to halt a City counter on the halfway line.
City responded to the dismissal by opting to keep the ball rather than plough forward in search of a tie-clinching goal, but space inevitably began to open up as United showed more ambition and became equally desperate.
However, while a second City goal - this time through Aguero - was ruled out for offside, the home side saw out the remaining minutes without much alarm as United ran out of steam.
Nevertheless, as they watched their neighbours celebrate yet another appearance at Wembley, United and Ole Gunnar Solskjaer will take heart from a second win at the Etihad Stadium in under two months.
MANCHESTER CITY (3-1-3-3): Bravo; Walker, Otamendi, Cancelo; Rodri; De Bruyne (Stones 90+1), Gundogan, Bernardo; Mahrez (D.Silva 68'), Aguero (Jesus 90+3'), Sterling
MANCHESTER UNITED (3-1-4-2): De Gea; Lindelof, Maguire, Shaw (Mata 79'); Wan-Bissaka, Fred, Matic, Williams; Lingard (Pereira 65'); Greenwood (James 46'), Martial