The centre-back had a deal at the Mestalla until June 2015, but the Portugal international - who made 20 La Liga appearances last season - has left the club for pastures new.
"I'm leaving Valencia in good conscience and with the affection of the fans," Costa told the club's official website.
"[These] have been years of great happiness where I felt the support of the fans in good times and bad times.
"I want to send greetings to all Valencianistas, the workers, my colleagues, the media and everyone with whom I have shared moments.
"It is time to start a new stage in my career, but I always take Valencia in my heart. I was proud to be captain of this team."
Costa was not thought to feature in the plans of new head coach Nuno Espirito Santo, who replaced Juan Antonio Pizzi earlier in July.
The 33-year-old has plied his trade at the Mestalla since May 2010 after joining the club from Wolfsburg.
His departure follows the sale of fellow defender Juan Bernat to Bayern Munich, while Jeremy Mathieu is set to be unveiled as a Barcelona player on Tuesday.