Speaking at a televised news conference in Barcelona after the attack, Joaquim Forn said: "We can confirm 13 dead and more than 50 injured."
The El Pais newspaper said Daesh had claimed responsibility for the attack through its media outlet called AMAQ.
Earlier, the paper cited police sources who confirmed it was a terrorist attack.
El Pais also added that a suspect had been identified -- Driss Oukabir, who according to his profile on social media networks is originally from Marseille, France, and resides in Ripoll, Spain. He allegedly rented the van.
Another newspaper El Mundo quoting police sources said two people, including Oukabir had been arrested in connection with the attack.
Radio Television Espanola said on Twitter that police were trying to negotiate with two suspects holed up in a bar in Barcelona following the attack.
Earlier, in a series of tweets on its official account, police asked people not to travel through areas around Las Ramblas street and Plaza Catalunya.
“Out of respect to the victims and their families, please, do not share images of injured in a trampling of #Ramblas of Barcelona,” they said.
At the time of the attack, Las Ramblas, a street in central Barcelona popular with both tourists and locals, was crowded with people.
In a written statement, the Turkish Foreign Ministry condemned the attack.
"We condemn in the strongest terms the terrorist attack perpetrated in Barcelona today," it said.
"We extend our deepest condolences and wish fortitude to the families of those who lost their lives and speedy recovery to those wounded in these heinous attacks which display once again the brutal face of terrorism and target all humanity.
"We reiterate our resolve to maintain our solidarity with Spain, which is a friend and an ally of Turkey, and to provide all kinds of support to this end."
U.S. President Donald Trump condemned the attack on Twitter, saying the U.S. "will do whatever is necessary to help".
"Be tough & strong, we love you!" he wrote.
Trump's top diplomat, Rex Tillerson, said the incident has the hallmarks "of yet another terrorist attack".
"We offer our condolences to the loss of life and the injuries that have occurred to so many innocent people yet again," he said.
"Terrorists around the world should know the United States and our allies are resolved to find you and bring you to justice."
Italian Prime Minister Paolo Gentiloni also expressed solidarity with the victims.
"Beautiful and friendly Barcelona is under attack. Italy close to local and Spanish authorities. Thoughts addressed to people affected," he tweeted.