Karake, 54, director general of the African country's National Intelligence and Security Services, was arrested at London Heathrow Airport on Saturday, British police said. He was remanded in custody to re-appear at the court on Thursday.
He is accused of ordering massacres while head of military intelligence in the wake of the 1994 Rwanda genocide.
And he is also accused of ordering the killing of three Spaniards who were working for the NGO, Medicos del Mundo.
Rwanda is seeking an explanation from Britain about the arrest, a minister said in remarks published on Tuesday.
"Western solidarity in demeaning Africans is unacceptable!!" Foreign Minister Louise Mushikiwabo said on Twitter, calling the arrest "lunacy."
Western solidarity in demeaning Africans is unacceptable!! It is an outrage to arrest #Rwanda-n official based on pro-genocidaires lunacy!— Louise Mushikiwabo (@LMushikiwabo) June 23, 2015
"We are handling the matter with the UK government," Justice Minister Johnston Busingye said in remarks published by the New Times newspaper on its website. "We will contest in the courts. We have sought explanation from the UK on this matter as well."
Rwanda's ambassador in London, William Nkurunziza, told the BBC that the charges levelled against Karake since 2008 in Spain were "politically motivated".
"We take strong exception to the suggestion that he's being arrested on war crimes," he told the BBC World Service.
He said the Spanish indictment had been criticised as highly politicised by many, including US diplomats.
In 2008, a Spanish High Court judge, Fernando Andreu, accused 40 Rwandan military and political leaders, including Karake, of engaging in reprisal killings after the country's 1994 genocide.
The judge indicted the officials for genocide, crimes against humanity and terrorism that resulted in the deaths of hundreds of thousands of civilians, including Spaniards.
In the genocide, about 800,000 people were slaughtered, mostly members of the Rwanda's Tutsi minority as well as moderates in the Hutu majority.
But the 2008 warrants were issued under Spain's doctrine of universal jurisdiction, which gave national courts the right to probe cases of grave human rights abuses committed abroad.
Spain under the ruling conservative government has since curbed this doctrine, and the judicial source said Tuesday that allegations against Karake concerning genocide, crimes against humanity and war crimes had been closed.
The case remained open, however, for the murders of eight Spaniards in Rwanda.
"The case is still open for crimes of terrorism," the source added. "This official is wanted in connection with what happened to several Spanish victims."
In a statement the British High Commission in Rwanda said the arrest was a "legal obligation" carried out on behalf of the Spanish authorities but stressed the UK had a "close relationship" with the Rwandan government.
"We co-operate closely on a growing range of regional and international issues," the statement said.
"Her Majesty's Government greatly values the close relationship with Rwanda and is committed to that relationship for the long term."
Jordi Palou-Loverdos, a lawyer representing nine Spanish massacre victims, told BBC's Newsnight on Tuesday: "We hope in the name of the victims that this time justice will be provided and Karenzi Karake will soon be delivered to the Spanish court to have a fair trial, where he can defend himself. And we hope that political or other interests will not neutralise the place for justice, truth and reparation."