However, even without the rested Cristiano Ronaldo, Real maintained their six-point lead at the top of La Liga as Mariano Diaz equalised six minutes from time before Ramos's late intervention.
Although Alvaro Morata's fine strike just after the break put Madrid ahead, Deportivo looked likely to come away with all three points when Joselu scored a quick-fire double against his former team.
"What we're achieving is down to these players".
The late fightback ensured Zidane's men have now surpassed Real's previous record mark of 34 games unbeaten from the 1988/89 season under Dutch coach Leo Beenhakker.
Real, bidding to win the Club World Cup for the second time in three years, beat Deportivo La Coruna 3-2 on Saturday thanks to a stoppage-time victor by captain Sergio Ramos, setting a club record 35 games unbeaten and keeping them six points clear at the top of La Liga. We made a few errors but that can always happen. "We will do some more research about them in the next few days". "We suffered, but we believed; it's awesome".
"The good thing is that the players are aware that, in addition to hard work, having a positive outlook can help us out".
Zidane also paid tribute to Ramos, who followed his 90th-minute leveller against Barcelona last week by powering in the victor deep in stoppage time.
"I am happy for all the players because they played great", said Zidane. "You can try to stop him but it's very hard".
"It wasn't a great performance".
"In the Bernabeu there is more added time, until Real Madrid score the final whistle doesn't blow". Real Madrid and Mexico's Club America will play in the December 15 semifinal of the FIFA Club World Cup soccer tournament.