Hazard was forced off in the second half of the 2-2 draw at the Santiago Bernabeu with Real coach Zinedine Zidane saying afterwards the Belgian had twisted his ankle.
But Hazard is expected to return within two weeks, meaning he should be fit for the first Clasico of the season against Barcelona at Camp Nou on December 18.
“After the tests performed on our player Eden Hazard by the Real Madrid medical services he has been diagnosed with bruising in his right leg,” Real Madrid said in a statement.
Hazard will miss Madrid’s game away at Alaves in La Liga on Saturday but could be back for their last Champions League group match against Club Brugge on December 11.
In between, Madrid play at home to Espanyol in the league on December 7.
Brugge’s draw with Galatasaray on Tuesday means Zidane’s team are guaranteed to finish second in Group A, regardless of the result in their final fixture.