The Spanish star, 21, has joined on a four-year deal after impressing for Anderlecht last term under former City captain and current Burnley boss Vincent Kompany.
Gomez left Barcelona's youth set-up to join Borussia Dortmund in January 2018, and went on to spend two seasons back in Spain on loan at Huesca before moving to Belgium last summer.
He told City's official website: "I am incredibly proud and happy to have joined Manchester City.
"City are the best team in England and in Pep Guardiola I have a chance to learn and develop under the most outstanding manager in world football.
"To be able to be part of this club is a dream come true for me and something any young player would aspire to.
"The number of trophies City have won over the past few years has been incredible and the style of football the team plays under Pep is the most exciting in Europe.
"Playing for and being guided by Pep and his coaches is going to be very special.
"I've heard so much about the City fans and how passionate they are, so I can't wait to get started and to try and help the club achieve even more."
Gomez - who was named Anderlecht's player of the year after a season in which he scored seven goals and registered 15 assists, and has represented Spain at every level from under-16s through to under-21s - will wear the number 21 shirt at City, as his compatriot David Silva did for the club.
He said: "I was a City fan when I was a kid. When we were kids, my brother used to support City. He's the main reason I became a fan as well.
"Once he was on a school trip to London and I asked him to bring a David Silva shirt. At that time, he was wearing the number 21 shirt and he was my favourite player because I used to play in his position when I was a kid.
"I feel very proud that I can wear his number, like Ferran Torres did here."
City's director of football Txiki Begiristain said: "We are delighted to be able to welcome Sergio to Manchester City. He is a very exciting young talent.
"Sergio's development in recent seasons has been very impressive - he is an accomplished defender with great potential.
"Over the past year, he has gained excellent experience playing in the Belgian League and in Europe with Anderlecht.
"We're sure that he will bring additional quality to our squad both in defence and going forward and that he will only continue to develop and improve playing under Pep and his coaches."