The 24-year-old, a Spain international, has made 170 senior appearances for his hometown team, and won the Europa League in his first full professional campaign, in 2015, and again the following season.
He joins the Cottagers as their eighth signing of the transfer window and will link up with the west London club as they return to the Premier League following last season’s Championship play-off final win against Aston Villa.
“I’m really pleased to have signed for the oldest club in the Premier League,” Rico told the club's official website following the confirmation of his move to Craven Cottage.
“I’m excited to start training, to meet all my teammates and, above all, to enjoy the season that starts this Saturday.”
Vice-chairman Tony Khan added that Rico’s arrival gives the club added depth as they look to make a success of their return to the top-flight of English football.
“Sergio is a world class goalkeeper, and thus, his services have been in great demand,” he said. “(He) joins a very talented cast of goalkeepers under contract at FFC, and he answers our mandate of making our squad the strongest and deepest it can be as we face top Premier League opposition.”
Rico stepped up from Sevilla B into their first team in 2014 and was in goal for their 3-2 Europa League Final win against Dnipro Dnipropetrovsk in 2015 and their 3-1 defeat of Liverpool in the same event one year later in 2016.
He made his international debut for Spain in a 6-1 drubbing of South Korea and was included in the national squad for the 2016 European Championship.
Fulham have also acquired the services of left-back Joe Bryan from Bristol City on a £6 million ($8m), with the defender having signed a four-year deal.
Bryan was widely linked with a move to Championship club Aston Villa but instead will join up with Slavisa Jokanovic’s side.
“Playing against Fulham the last couple of years, the style of play has really stood out to me and it will suit me," he told the club's website.
“The opportunity to play in the Premier League is one you can’t turn down as you may never get it again in your lifetime. I’m used to playing fast attacking football, so hopefully I can slot right in.”
Fulham begin their Premier League campaign with back-to-back London derbies, firstly against Crystal Palace at Craven Cottage on August 11 before they travel to Wembley to face Tottenham a week later on August 18.