Matic’s first goal in more than a year offered United the tantalising possibility of turning round the tie after City had made the early running and twice forced saves out of United goalkeeper David de Gea.
Raheem Sterling had the ball in the net six minutes after United went ahead but was offside and 13 minutes into the second half he conjured up a horror miss as he tried to round de Gea.
The goalkeeper did enough to force him a touch wide, allowing time for defenders to drop behind and when Sterling found space to strike, he was off balance and hit the ball over the crossbar.
There was another let-off for United when Harry Maguire was caught in possession, but City again dithered and allowed the defence to block the chance.
With six minutes to play, Sergio Aguero netted after a clever through ball from Joao Cancelo, but was marginally offside.
United had a rare opportunity to get the second goal with an 88th-minute free kick after Maguire was fouled just outside the penalty area, but Fred struck his shot against the wall.
City boss Pep Guardiola said over 180 minutes his side had been better than United.
“We are happy to go to Wembley for our third final in a row, and we will fight for the competition like we fought three years ago,” he told Sky Sports.
“We didn’t convert them [chances] and of course have to learn from that and we have to be more clinical.”
The final is on March 1 when City will seek to win the trophy for a third successive year and a fourth time in five seasons.
Defeat in the tie ended United’s hopes of winning the trophy in what has been a disappointing season, but manager Ole Gunnar Solskjaer was happy enough after a second successive win at the Etihad.
“I am so proud of these players. They have beaten City twice at their ground. They have come so far these players. They have given me absolutely everything,” said Solskjaer.
“Even though we did not create too many clear cut chances we pressed them back. It pleased me.”