The Premier League champions will take a slim lead to the Spanish capital next week wondering what could have been.
City were ahead inside the first two minutes, with Kevin De Bruyne scoring the fastest goal ever in Champions League semi-finals history. Gabriel Jesus added a second after 11 minutes but Karim Benzema halved the deficit before halftime.
Phil Foden headed City further ahead early in the second half but Vinicius Junior reduced the deficit straight away with a piece of brilliant individual magic before Bernardo Silva restored the hosts' two-goal advantage.
Benzema added his second from the spot with a panenka penalty to give Madrid hope heading into the second leg at the Santiago Bernabeu next week.
At times, City could have been out of sight but were guilty of some woeful finishing as well as some selfish play from the forwards.
Foden has told his side they will need to be more clinical in Madrid if they are to qualify for the final which will take place in Paris next month.
"For the fans watching, it was obviously a great game of football," he told BT Sport.
"We are playing a side that have won the Champions League many times and if we give the ball away, they are going to punish us.
"It is something we need to work on in the second leg. The tie is still on.
"It was a great game; end to end. We started really well and could have killed them off. In these games, we need to take more chances."
Nonetheless, City boss Pep Guardiola was content with how his side performed against the record European champions.
"We played a fantastic game against an incredible team," the head coach added.
"All around the world and for Manchester City, we are so proud. But it is about reaching the final and sometimes, football happens.
"We go to Madrid to try to win the game. Both teams want to attack and have the quality to play.
"Congratulations to Carlo and his team as they are so good. At the same time, we saw ourselves that we can be there."
