Los Blancos headed to Stamford Bridge after a 1-1 draw in the first leg, but were easily beaten 2-0 by the Blues, who could realistically have doubled their tally on the night.
Hazard, meanwhile, was as ineffective as most of the rest of his team-mates, so drew the ire of some for his lacklustre showing against his old side - but it was his post-match antics which upset others.
After full time he was pictured talking and joking with former team-mates, with a vocal few on social media calling for him to be sold and the Spanish media going all-in on their disappointment of the Belgian, with TV show El Chiringuito front and centre among them.
One clip circulating on social media showed presenter Josep Pedrerol venting at Hazard - complete with damning background music and sighs of seriousness before he began talking - suggesting that he was “laughing” at Real Madrid supporters and didn’t deserve to continue at the club, comparing his perceived lack of commitment to the cause to that of Gareth Bale.
Hazard himself has responded with a post on his Instagram story, writing a short message of apology.
“I am sorry. I have read lots of opinions about me today and it was not my intention to offend any Real Madrid fans,” Hazard wrote.
“It has always been my dream to play for Real Madrid and I came here to win. The season is not over and together we must battle for La Liga! Hala Madrid!”
After their exit from Europe, Real are now in a three-way battle for the title in Spain.
They sit second, two points behind city rivals Atletico Madrid and level on points with third-place Barcelona. Sevilla are four points back, with their own title hopes hit by a defeat last time out.
All four clubs have four matches remaining, with no further cup or Continental distractions for any of them.
Barcelona are at home to Atletico on Saturday afternoon in one pivotal clash, before Real host Sevilla on Sunday night in another.
Hazard has made just seven league starts this season due to several injuries, scoring twice and claiming one assist.
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