The whole group arrived on three separate boats, after some were transferred to two Italian vessels to make the journey safer. It transported 274 people.
The Aquarius arrived carrying 106 people rescued from unstable boats near Libya. Aid workers awaiting their arrival clapped and cheered as the first passengers walked down the gangway.
However, 42 migrants rescued this week by a U.S. ship off Libya before being transferred to an Italian coastguard vessel already carrying 500 migrants would be allowed to land in Italy, media reported in Rome on Sunday.
High waves and winds had forced the convoy to take a detour on its 1,500-kilometer (930-mile) voyage to Spain.
Salvini refused to permit the migrants saved by the Aquarius to land in Italy.
Italy's right-wing interior minister says he hopes Spain accepts "66,629" more migrants.
The refusal by Italy and Malta to allow the Aquarius to enter their ports has created a row between European Union members over how to handle immigration.
The passengers were greeted by more than 2,300 Red Cross volunteers and other humanitarian workers, including police, doctors and translators.
The first person through the process was a 29-year-old man from South Sudan.
"They also look exhausted", she told AFP.
"It has been a very emotional week, mixed emotions, very positive emotions due to all the people we met", the Spanish photographer said.More news: LA Mayor Eric Garcetti to visit Texas detention center, condemn family separation
The 270 migrants on board soon began to disembark after medical staff had made a preliminary inspection.
"The disgusting stories that we hear from people who come out of Libya is gut-wrenching", he said.
"We celebrate that rescued people are finally in a safe place, where they will be assisted with dignity", he said.
Spanish authorities are examining the migrants on a case-by-case basis to see who may qualify for asylum.
"They will be dealt with according to our laws, without exception", Spain's minister of public works, Jose Luis Abalos, said Saturday.
Spain eventually stepped in and agreed to receive the refugees as a "political gesture" to "oblige Europe to forge a common policy to a common problem", its Foreign Minister Josep Borrell said.
"I left Sierra Leone nearly 11 months ago...because I want to protect my baby girls who they want to circumcise". Another Italian navy ship with the rest of the migrants from the Aquarius will arrive later.
The destination wasn't immediately announced, but the coast guard said it wouldn't dock until at least Tuesday. "Finally the Germans, the French, the Belgians, the Dutch and even the Spanish are listening to us", said Salvini, who is also deputy premier and leader of the League party.
Although their specific number has not been given, the Nigerians were among the 31 nationalities rescued from the Mediterranean Sea by the Aquarius.
Some 600,000 rescued migrants have arrived in Italy in recent years. Italy's new government will make any compromises on migration policy even more hard.
The warmer weather has caused a spike in migrants taking off from North Africa for Europe. It also recovered four bodies and declared another 43 people missing at sea.
The United Nations refugee agency is warning against a repeat of the standoff that left a private rescue ship carrying 629 migrants at sea for a week.