Lucas Vázquez put the La Liga champions ahead just 57 seconds into the match, before Kylian Mbappé scored with a superb finish, assisted by Jude Bellingham.
Rodrygo made it 3-0 shortly after the second half began, but Carlos Benavídez pulled one back for the visitors in the 85th minute.
Deportivo’s Kike García added a second goal just a minute later, creating a tense finish, but Carlo Ancelotti’s team managed to hold on.
Mbappé’s goal brings his total to seven in nine appearances this season for Real Madrid, with six coming in his last seven games.
He and Bellingham also linked up in the 23rd minute, but the Frenchman’s effort into the bottom corner was disallowed for offside.
Mbappé joined Real Madrid in July, signing a five-year contract after leaving Paris Saint-Germain.