The Belgium international has been blighted by ankle and leg problems since joining Zinedine Zidane's Spanish giant from Chelsea nearly 18 months ago.
It was not immediately clear what his latest problem was, with Hazard able to walk off the Estadio Alfredo Di Stefano pitch – albeit in obvious discomfort – to be replaced by Rodrygo after 28 minutes of the game.
On Tuesday, though, Madrid confirmed he has sustained an injury to the rectus femoris muscle in his right thigh.
No timescale has been put on his return but he has not been named in Madrid's squad for Wednesday's (AEDT) Champions League trip to Shakhtar Donetsk.
Saturday was just Hazard's third top-flight appearance of the season, after previous outings against Villarreal and Huesca.
He scored in the 4-1 victory over Huesca but has failed to create a single chance for his team-mates in his 153 minutes on the pitch this term in LaLiga.
Last season he started just 14 games in the league, playing a minor role as Madrid won the title in his first campaign at the club.