The Blues, who lost to Arsenal in last season's final, will take on the winner of the second semi-final between Leicester City and Southampton for the trophy on May 15.
Man City's quadruple bid is now over, but Pep Guardiola's side can still win the EFL Cup, Champions League and Premier League in the coming weeks.
Chelsea had the ball in the back of the Man City net early on through Ziyech, but the effort was ruled out for offside, and a largely uneventful first period finished goalless.
Thomas Tuchel's side had the better of the opening 45 minutes, but opportunities were few and far between in the English capital, with both teams struggling for creation in the final third of the field.
Chelsea made the breakthrough early in the second half, though, when Timo Werner raced clear before delivering a low cross into Ziyech, who could not pass up the chance to convert from close range.
Ziyech had a golden chance to score a second four minutes later, but the 28-year-old fired straight into the body of Zack Steffen, before Ruben Dias headed a brilliant opportunity for Man City over the crossbar as the second period progressed.
The Citizens continued to move the ball in the latter stages as they searched for a leveller, but Tuchel's side, who had another goal ruled out late on for offside, held on to advance to the final of the FA Cup.
Man City will resume their Premier League campaign away to Aston Villa on Wednesday night, while Chelsea are next in league action at home to Brighton & Hove Albion on Tuesday evening.
6th min: Chelsea have the ball in the back of the Man City net as Werner feeds Ziyech, who fires past Steffen, but the goal is correctly ruled out for offside.
33rd min: Werner uses his pace to break into the Man City box, but Aymeric Laporte is across to make a brilliant block; James then fires wide of the post from the resulting corner kick.
51st min: Half-chance for the Citizens as a dangerous cross from Phil Foden finds Jesus, who heads towards Sterling, but James does well to make the block before Kepa collects the loose ball.
55th min: Chelsea make the breakthrough in the 55th minute of the contest as Ziyech converts from close range after Werner had picked out his teammate with a low cross.
59th min: Huge opportunity for Chelsea to score a second as Ziyech breaks into a one-on-one situation following a mistake from Dias, but Steffen makes a smart save to prevent the former Ajax playmaker from finding the back of the net for a second time.
69th min: Massive chance for Man City as Rodri heads towards Dias inside the Chelsea box, but the centre-back somehow nods over the Chelsea crossbar from a very dangerous position.
77th min: Werner races the length of the Man City half before attempting to pick out the bottom corner, but Steffen is on hand to make a relatively routine save.
82nd min: Smart save from Kepa to keep out a deflected strike from Rodri as Man City continue to move the ball in the final third of the field.
93rd min: Christian Pulisic has the ball in the back of the net for Chelsea, but the attacker's effort is ruled out for offside, cutting the celebrations short.
94th min: Smart save from Kepa to keep out a header from Rodri as Man City desperately attempt to take the FA Cup semi-final into an additional 30 minutes.
MAN OF THE MATCH - N'GOLO KANTE
Kante's performance for Chelsea was outstanding, and there is no question that the France international played an important role in his side advancing to the final of the FA Cup.
The midfielder's energy and work rate was infectious for the Blues, who defended excellently over the course of the 90 minutes, limiting Man City to very little in the final third of the field.
Kante has had his injury problems this term, but the 30-year-old demonstrated that he is still one of the leading midfielders in world football with his display at Wembley.
De Bruyne was forced off the field in the early stages of the second period, having seemingly picked up an ankle problem, which will be a huge concern for Man City at this stage of the season.
The Citizens are now out of the FA Cup but are still competing on three fronts and will not want to take on Paris Saint-Germain in the Champions League without arguably their most important player.
Most FA Cup final appearances since Roman Abramovich took over Chelsea:— William Hill (@WilliamHill) April 17, 2021
🔵 Chelsea (8)
🔴 Arsenal (5)
🔴 Man Utd (5)
🔴 Liverpool (2)
🔵 Man City (2)
🔵 Portsmouth (2)
Blue is the colour. #FACup pic.twitter.com/4Dq4QFAbHV
Possession: Chelsea 45%-55% Man City
Shots: Chelsea 5-11 Man City
Shots on target: Chelsea 3-3 Man City
Corners: Chelsea 3-6 Man City
Fouls: Chelsea 8-12 Man City
CHELSEA (3-4-3): Kepa; Azpilicueta, Silva (Zouma 88'), Rudiger; James, Kante, Jorginho, Chilwell; Mount (Pulisic 70'), Werner (Havertz 79'), Ziyech (Emerson 79')
MAN CITY (4-2-3-1): Steffen; Cancelo, Laporte, Dias, Mendy; Fernandinho, Rodri; Torres (Gundogan 64'), De Bruyne (Foden 48'), Sterling; Jesus