"A minor was carried away by the sea on the Rapadoira breakwater in Foz," a town on the coast of the northwestern Galicia region, the national coastguard said in a message on Twitter.
Coastguards and police were searching for him, it added.
Spanish news agencies said the boy was aged 15 and was swept away as he was cycling along the waterfront with a friend who escaped.
Huge waves pounded the coasts, and mechanical diggers and emergency workers laboured with their hands to shift lumps of concrete and other debris caused by the storm.
The tidal storms struck several northern regions, Spanish media reported.
In the Basque Country, huge waves broke over the seafront to flood the major tourist city of San Sebastian, as captured in the video below.
Other videos showed rivers of seawater gushing through the streets, causing as-yet untold damage to shops and residential buildings.