“Eleven months ago we had no representation in any government institution. Today we are the third political force in Spain,” Mr Abascal told his supporters at a victory rally on Sunday night, according to El País.
“We have told the left that they have no moral superiority.”
The Socialist PSOE party, which won the most seats, will be the first to try form a government, but it will be a challenging task because neither left nor right-wing parties collectively won enough seats for a majority.
Pablo Iglesias, leader of the left-wing Podemos, said the election had served only to “strengthen the right and turn the far right into one of the strongest in Europe.”
Mr Abascal has frequently referenced fascist ideology and expressed sympathy with Spain’s former dictator Francisco Franco during his campaigns.
He has promised to outlaw separatist movements including parties in the northeastern Catalonia region and to crack down on immigration, particularly among Muslims, speaking of a Reconquista — a nod to the 15th century Spanish Inquisition.