The law allowing dual citizenship for descendants of Jews who were forced to flee Spain in 1492, known as Sephardim, was approved by the Spanish parliament in June and came into force on Thursday.
But Justice Minister Rafael Catala said on Friday that the government had approved an extra measure to save paperwork for thousands who had already filed applications for nationality before the latest law came into force.
The new decree "has allowed us at one stroke to grant nationality to 4,302 people of Sephardic origin" whose applications under previous legislation were already pending, Catala told a news conference after a cabinet meeting.
"This is one more step in developing the law for granting nationality to the Sephardim," he said.
"It seemed fair, rather than making them go through the process of filing their applications again, to speed up the process."
The measure aims to correct what the Spanish government has called the "historic mistake" of the country's Catholic monarchs sending Jews into exile in 1492.
Jewish groups welcomed the law in June, though some Jewish leaders complained the requirements are too burdensome.
Applicants do not have to be practising Jews but they must have their ancestry vetted by Jewish authorities and prove a "special connection" to and knowledge of Spain.
Historians believe at least 200,000 Jews lived in Spain before the monarchs Isabella and Ferdinand ordered them to convert to the Catholic faith or leave the country on pain of death.
Up to 3.5 million people around the world are thought to have Sephardic Jewish ancestry.