The top-ranked Japanese had to fight for every point against the determined Christinna Pedersen and Kamilla Rytter Juhl and dug themselves out of a huge hole to win an 18-21 21-9 21-19 thriller at the Riocentro.
Trailing 19-16 in the final minutes, Matsutomo and Takahashi won five successive points to crush Denmark's hopes of a first badminton title in 20 years.
Matsutomo and Takahashi went one better than compatriots Mizuki Fujii and Reika Kakiiwa who won the silver in the event at London.