Madrid trailed at half-time but Benzema netted a towering header to restore parity approaching the hour-mark before the 31-year-old repeated the feat with nine minutes remaining.
"I am very proud to play here at the Bernabéu", he said.
Zidane made it clear he was unhappy with his side's first-half performance, adding: "We had to change our performance because we could not continue like in the first half".
A wayward shot from outside of the box, coupled with several unsuccessful passes, then brought further opprobrium for a player who, in the wake of Cristiano Ronaldo's departure, has struggled to help fill the void left by the Portuguese.
Meanwhile, Madrid boss Zinedine Zidane refused to reveal his plans on the superstar, who memorably scored one of the winning goals in the Merengues' 4-1 rout of Liverpool in the 2018 Champions League finals.
"The first half we lacked confidence, above all at the start of the game, which was very hard".
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"The situation is hard", he said.
"In this situation, we will go out little by little, insisting on working and thinking positively", he said. The second half they upped things today. Zidane's post-match admission that "maybe in these games we lack some motivation" was hardly an inspiring call to arms.
ZINEDINE ZIDANE refused to answer questions about Gareth Bale's future after the player was jeered by fans at the Bernabeu on Saturday.
Hazard, 28, has a year left in his deal with the Stamford Bridge outfit and can leave for nothing in a just over a year when his Stamford Bridge contract expires.