The Blues splashed £60million on the player and pipped rivals Manchester City to his signature.
The 24-year-old follows previous summer recruits Raheem Sterling, Kalidou Koulibaly and Carney Chukwuemeka into Stamford Bridge.
Manager Thomas Tuchel was delighted with the deal, saying: "He's young, hungry, mobile and a very intelligent player.
"It helps in depth and in quality, and it helps with Ben, to escape the pressure of I have to deliver and we need you absolutely now.
"We have of course at the moment in this position Marcos Alonso, Kenedy and Emerson.
"But hopefully when things go through Marc is the option on this position.
"And I think he can play very well in the back three, so it's a bit of a profile of Azpi (Azpilicueta) on the left side. He gives us many options."
Chelsea announced the deal by mocking Brighton's social media post from a couple of days prior, in which the club stated that the deal was not completed.
Cucurella's arrival hands Tuchel the kind of depth and quality at wing-back he has craved ever since his first days at Stamford Bridge.
The player said: "I'm really happy, it's a big opportunity for me to join one of the best clubs in the world and I'm going to work hard to be happy here and help the team," said Cucurella.
Chelsea chairman and co-controlling owner Todd Boehly hailed another signing already boasting Premier League experience, one of the Blues' main aims this summer.
"Marc is an elite defender of proven Premier League quality and he further strengthens our squad going into the new season," said Boehly.
"We continue to work on and off the pitch, and we're delighted Marc will be a part of the present and future at Chelsea."