He also resigned from the Communist party's central committee and political bureau, effectively removing himself from political life in Cuba.
Perez Roque, the former foreign minister, said he would also quit the Council of State, the National Assembly and the party central committee.
"I fully recognise that I committed errors that were broadly analysed in a meeting [with the political bureau]. I assume my full responsibility for them," he said in the letter, also dated on Tuesday.
|Perez Roque, left, Lage were former allies of
Fidel Castro [AFP]
At least 20 officials were moved, demoted or promoted by Raul Castro, the Cuban president, on Monday, in a move the government said was intended to make Cuba's government more compact and functional and to work towards "perfecting" the Cuban system.
In an apparent reference to Perez Roque and Lage, Fidel Castro said in an article on a government website on Tuesday the two had developed ambitions that led them to "an undignified role".
Castro, who resigned the Cuban presidency last year due to ill health, also said the men were removed as "the external enemy filled itself with expectations for them," although it was not clear who this referred to.
Perez Roque, who had been Havana's chief diplomat since May 1999, was replaced by Bruno Rodriguez, his deputy.
And Lage was replaced as cabinet secretary by General Jose Amado Ricardo Guerra, a former top military official.
Lage had been credited with helping to save Cuba's economy by implementing economic reforms after aid from the Soviet Union ended in the early 1990s, while Perez Roque was once personal secretary to Fidel and a former leader of the Communist party's youth organisation.