The Belgian was playing in only his second match since returning from an ankle injury, which kept him out of action for almost three months.
Real's best chance against Levante fell to Hazard, who scuffed the ball when he had a clear sight of goal early in the second half. He limped off shortly afterwards and was replaced by Vinicius Junior.
Levante forward Jose Luis Morales won the game with an unstoppable strike in the 79th minute.
Hazard's injury comes at the beginning of an important week in Real's season.
They host Man City at the Bernabeu on Wednesday in the first leg of their Champions League last-16 tie.
Barcelona then travel to the Bernabeu on Saturday for El Clasico.
Real fall to second place in La Liga on 53 points after 25 games.
Barcelona are the new leaders at the top of the table with 55 points.