The band said they were "shocked and filled with overwhelming sadness with the untimely passing of our dear friend, family member, and bandmate".
The statement went on to say that the keyboard player had a "heart of gold" and their "hearts are with his family".
Fletcher, who was 60, was a founding member of the band which found success with their dark electronic sound with hits including Personal Jesus.
Depeche Mode had chart success in the 1980s and 1990s with singles including New Life, Enjoy the Silence and Just Can't Get Enough.
The group was founded in Basildon in 1980 and was inducted into the Rock and Roll Hall of Fame two years ago.
In a statement posted on social media, the band said: "Fletch had a true heart of gold and was always there when you needed support, a lively conversation, a good laugh or a cold pint."
Fletcher, who was born in Nottingham, remained with the band throughout its four-decade history, which included two number one albums, Ultra and Songs of Faith and Devotion.
His last studio album with the band was 2017's Spirit, their 14th, which reached number five in the UK albums chart.
He leaves behind wife Grainne, to whom he was married for almost 30 years, and two children, Megan and Joe.
Lol Tolhurst, drummer with rock band The Cure, described the news of Fletcher's death as "very sad".
"I knew Andy and considered him a friend. We crossed many of the same pathways as younger men," he tweeted.
In a post on Facebook, synth-pop duo Pet Shop Boys said: "Fletch was a warm, friendly and funny person who loved electronic music and could also give sensible advice about the music business."
Sister Bliss, from electronic band Faithless, described Fletcher as a "keyboard warrior" and tweeted that he was "gone way too soon".
Dave Gahan (left) and Andrew Fletcher of Depeche Mode perform live on stage at the O2 Arena in London
A picture of Andy Fletcher is projected on a big screen as Depeche Mode perform in Anaheim, California
Andy Fletcher performing in Rome in 2013