Karim Benzema had given Madrid an early lead at the Bernabeu before Hazard found the net, for the first time since his £89million move from Chelsea, in first-half stoppage time.
It was a delightful finish from the Belgian, who was under increasing pressure to find the net following a string of under-par performances for Los Blancos.
Luka Modric extended the advantage after the break before Darwin Machis pulled a goal back from the penalty spot.
Domingos Duarte then scored again for the visitors with 12 minutes to make Madrid sweat but James Rodriguez made sure of victory three minutes into stoppage time.
Real got off the perfect start as Benzema put them ahead with just two minutes gone.
Gareth Bale’s cross from the right picked out the France international at the back post and Benzema fired home first time into the net from close range.
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