Two second half goals from John Stones and Fernandinho gave Pep Guardiola’s side victory in the match.
Manchester City will now meet Jose Mourinho’s Tottenham Hotspur at Wembley in the April 25 final as they seek their fourth League Cup triumph in a row.
In a frantic opening to the game, both teams had efforts ruled out for offside.
Stones turned the ball into his own net but he was saved by the flag.
Also, Ilkay Gundogan converted a low ball in from Phil Foden but, again, the attempt was rightly ruled out.
Kevin De Bruyne struck the post with a thundering drive from outside the penalty box.
Later on, Foden also put the ball in the net for Manchester City but the visitors were foiled by offside once again.
It was an open and entertaining game, with Manchester City looking sharper.
But Manchester United were competitive as they sought revenge for last year’s defeat by their neighbours at the same stage of the competition.
Manchester City then went ahead five minutes after the break when Foden whipped in a free-kick from the left through a crowded box.
Defender Stones bundled a cross home at the back post, the ball going in off his thigh.
Manchester United goalkeeper Dean Henderson produced a brilliant save to tip over a strike from Riyad Mahrez, after the Algerian had burst forward from the halfway line.
Fernandinho made sure of the win seven minutes from time with a stunning volley as he pounced on a headed clearance from Aaron Wan-Bissaka.
Manchester United have now lost at the semi-final stage in their last four Cup campaigns, leaving manager Ole Gunnar Solskjaer still searching for his first final since taking over at the club.