Real said Hazard, 28, had signed a five-year contract.
His contract at the Premier League club was due to expire next year but the Blues drove a hard bargain for the forward, who will reportedly cost Madrid an initial €100 million with another €45 million in potential add-ons.
The full amount would trump the €101 million Madrid paid Tottenham for Gareth Bale and the €91 million they spent to sign Cristiano Ronaldo from Manchester United.
An unveiling at the Santiago Bernabeu is expected next week, once Hazard returns from international duty with Belgium, who play Euro 2020 qualifiers against Kazakhstan on Saturday and Scotland on Tuesday.
It means Madrid's spending spree is picking up pace following the arrivals of defender Eder Militao and striker Luka Jovic from Porto and Eintracht Frankfurt respectively.
But Hazard is the gem Zinedine Zidane can build his new team around and brings the kind of stardust Madrid fans have craved ever since the sale of Ronaldo to Juventus last summer.
Ronaldo's departure created a void up front that Madrid failed to fill last season, the team struggling for goals en route to finishing 19 points behind Barcelona in La Liga and crashing out to Ajax in the last 16 of the Champions League.
Hazard, who scored 110 goals in seven seasons at Chelsea, will be expected to help plug the gap has long-been admired by Zidane and Madrid president Florentino Perez.
"We have wanted to sign Hazard for several years and I hope he will come this year," Perez told Spanish radio station Onda Cero last month.
Less clear is where Hazard will fit into Zidane's starting line-up. Assuming he plays in his favoured position on the left of a front three, it would mean displacing the 18-year-old Brazilian, Vinicius Junior, who enjoyed a brilliant breakthrough year last season.
Chelsea said the club had tried to persuade Hazard to stay.
"Although it is with sadness we say goodbye to Eden - and we made it absolutely clear to him the club wished him to stay - we respect the decision he has made to take on a new challenge in a different country and follow his childhood dream of playing for Real Madrid," Chelsea club director Marina Granovskaia said in a statement.
Hazard's arrival also adds more uncertainty around the future of Gareth Bale, who has played both on the left and right of the attacking trio in recent months but is clearly not part of Zidane's plans.
Madrid want to sell Bale to raise money for more signings, with French left-back Ferland Mendy the next player expected to arrive, from Lyon.
But the Wales international will not be forced out and has so far shown no interest in the handful of clubs that have shown interest. Neither Bale nor his agent Jonathan Barnett have spoken to the club since the end of the season.
Hazard though would do well to match Bale's record in Madrid as he looks to add to the trophies he won at Chelsea, which included two Premier League and two Europa League titles, as well as one FA Cup and one League Cup.
At the end of the 2014-15 season, in which Chelsea won the league, Hazard was awarded both the PFA Players' Player of the Year and the Premier League Player of the Season. He was named in the PFA Premier League team of the year four times, in 2013, 2014, 2015 and 2017.
If Madrid pay the full amount, Chelsea stand to make a profit of more than €100 millionf, having signed Hazard for close to €40 million from Lille in 2012.
Chelsea had just won the Champions League when Hazard chose them over several other clubs but they have failed to replicate the success in the competition. After reaching the semi-finals in 2014, Chelsea have not gone past the last 16 in the last five seasons.