The 24-year-old caught the eye of the Premier League's top teams after a strong debut season on the South Coast, registering one goal and two assists in 38 matches across all competitions.
Cucurella started 35 Premier League matches under Graham Potter and was initially on the radar of Manchester City, who baulked at Brighton's £50m asking price.
As a result, Chelsea were able to hijack a move for the former Barcelona and Getafe man, with the Premier League champions refusing to meet the Seagulls' financial demands.
Earlier this week, Brighton denied the existence of an agreement with Chelsea for Cucurella, posting on Twitter: "Contrary to inaccurate reports from numerous media outlets this evening, no agreement has been reached with any club to sell Marc Cucurella."
The Blues proceeded to aim a sly dig at the Seagulls with their social media retort of the Spain international's signing, and the club have confirmed that he has penned a six-year deal until 2028.
"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," Cucurella told chelseafc.com.
Chelsea chairman Todd Boehly added: "Marc is an elite defender of proven Premier League quality and he further strengthens our squad going into the new season. 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."
"We're all looking forward to getting started on the pitch now, and Marc only adds to that feeling. We welcome him to the club!"
Brighton confirmed in their own statement that they would receive a club-record transfer fee for the sale of Cucurella, which is expected to be an initial £55m worth up to £62m with add-ons.
"We did not want to lose Marc, but he wanted the opportunity of playing Champions League football with a view to accelerating his progress to Spain's World Cup squad," Brighton head coach Potter told brighton&bovealbion.com.
"Of course we are disappointed to see Marc leave, but we wish him well for the future. He had an excellent first season in the Premier League, deservedly winning both our Player of the Season awards, and has been an absolute pleasure to work with over the past 12 months."
Cucurella - who is capable of playing as a left-back, left-wing back or as part of a three-man backline - becomes Chelsea's sixth signing of the summer window after Raheem Sterling, Kalidou Koulibaly, Gabriel Slonina, Carney Chukwuemeka and Omari Hutchinson.
Brighton announced the capture of Levi Colwill on a season-long loan from Chelsea just after confirming Cucurella's exit, with the 19-year-old arriving to replace the departed Spaniard in a separate transaction.
Chelsea manager Thomas Tuchel had already confirmed that a move for Cucurella was in the works during Friday's press conference, and the defender arrives to compete with Ben Chilwell for the left-wing back spot, as well as providing centre-back cover following Andreas Christensen and Antonio Rudiger's exits.