Madrid have signed Hazard on a five-year deal for a fee that could rise to €140million, although the deal is subject to the Belgium star passing a medical.
Hazard has long been linked with Madrid but confirmed his intention to leave the Blues after scoring twice as they beat Arsenal in last month's Europa League final.
And in a lengthy farewell posted to social media, Hazard explained it was a wrench to depart Stamford Bridge after seven years with Chelsea.
"To my Chelsea friends and family, you now know that I will be joining Real Madrid," Hazard wrote on Facebook.
"It's no secret that it was my dream to play for them since I was a young boy just scoring my first goal. I have tried my very best not to distract myself or the team through this difficult period of speculation and media attention, especially the last six months.
"Now the clubs have reached an agreement I hope you understand I had to pursue my next chapter, just as each and every one of you should when you have the chance to pursue your dreams. Leaving Chelsea is the biggest and toughest decision in my career to date.
"Now it's in the open I would like to put on record one thing was always clear to me, I have loved every moment at Chelsea and not once did I ever consider, nor would I have left for any other club.
"I was only 21 when I joined, so I have grown up as a man and a player with all of you, you have helped me become captain of the Belgium national team after all. There have of course been tough times, for the team as a whole and for me personally, this is professional football.
"However, football to me is about having a ball at my feet, playing games and enjoying every moment, we are lucky to play the beautiful game and this is always my advice to those who ask for a coaching tip! Play football and have fun!
"Even when we were not playing well, I tried to do my best and this has been thanks to your amazing support. That fighting spirit, desire and the refusal to ever give up is part of the culture here at Chelsea and it comes directly from you."
Hazard won a brace of Premier League titles with Chelsea, as well as one EFL Cup and one FA Cup.
He becomes the fourth new arrival at Madrid ahead of the 2019-20 season, following Eder Militao, Luka Jovic and Rodrygo Goes to the club.
Lyon left-back Ferland Mendy is widely expected to join Zinedine Zidane's squad rebuild at the Santiago Bernabeu, while stars including Paul Pogba and Neymar have also been linked.