Real, without the rested Cristiano Ronaldo, fought back from 2-1 down in the final six minutes to restore their six-point advantage over arch-rivals Barcelona, who won 3-0 at Osasuna earlier on Saturday.
Zinedine Zidane's side laboured in an uninspiring first half which saw Depor midfielder Celso Borges head against the post.
Striker Alvaro Morata's 20-yard drive shortly after the restart appeared to ease the tension among the home fans.
However, Depor substitute Joselu - on loan from English Premier League side Stoke - swung the match dramatically in favour of the visitors.
He powered in the equaliser with a rising drive into the top-left corner after on-loan Aston Villa midfielder Carles Gil dispossessed Casemiro on the edge of his penalty area.
And Joselu, who came through the ranks at the Bernabeu, increased the nervousness by pouncing on more defensive uncertainty to slot home.
But the drama did not end there.
Real equalised with six minutes left when substitute Mariano Diaz scored his first La Liga goal with a fortuitous effort that flew in off his shoulder.
Then, skipper Ramos - whose late equaliser earned Real a vital draw at Barcelona last weekend - headed in Toni Kroos' corner to spark jubilant celebrations at the Bernabeu.
It means that Zidane's men have surpassed Real's previous record of 34 games unbeaten, set under Dutch coach Leo Beenhakker in 1988-89.
- Relive Madrid's remarkable win against Deportivo
Madrid on course to end barren Liga run?
Madrid have not won La Liga since 2012 but will end the calendar year in pole position after rediscovering their resilience since club legend Zidane replaced Rafael Benitez in January.Los Merengues went into Saturday's game having taken 87 of the 102 points on offer in their previous 34 league matches and, although Zidane's side have not always been fluent, they have developed a useful knack of refusing to acknowledge when they are beaten.The French World Cup winner has instilled a never-say-die attitude in his side that has yielded some important points in the biggest moments.And, just seven days after Ramos salvaged a draw in El Clasico, that fighting spirit was apparent again against fifth-bottom Deportivo.Madrid were missing a number of stellar names as Zidane named a much-changed side with one eye on next week's Club World Cup in Japan.Without Ronaldo, Luka Modric and Karim Benzema, plus injured Wales forward Gareth Bale, they lacked fluency and looked defensively vulnerable.But the home side continued to believe they could turn the game back in their favour as Depor dropped deeper.The pressure eventually told. The visiting defence buckled as Ramos rose highest, the away players buckling to the turf in despair.