The royal couple sent their best wishes to the 76-year-old Argentinian pope who will be known in Spanish as Santo Padre Francisco I, reported ABC on Wednesday.
Spanish prime minister Mariano Rajoy also welcomed the news of the election of Jorge Mario Bergoglio.
In his message, Mariano Rajoy said he hoped to "strengthen the special relations between the Holy See and Spain".
"On this special, historic occasion in which the church is more universal than ever, I wish to demonstrate that my government is ready to maintain and strengthen these special relations," said Rajoy.
The new pope — whose choice came as a surprise to most observers — is the first pope from the South American continent and the first native Castellano-speaking pope since the Borjas of the 15th and 16th centuries.
As the child of Italian immigrants, Francis I is also a native Italian speaker.
Francis I is also the first member of the powerful Jesuit order to become pope.