He also resigned from the 2014 World Cup organising committee.
Mr Teixeira, who led the federation for 23 years, had come under pressure over corruption allegations, which he has denied.
He has been investigated on suspicion of crimes including tax evasion but has never been convicted of any offence.
Former Sao Paulo Governor Jose Maria Marin, 79, will be replacing Mr Teixeira in both posts.
Mr Marin read out the letter in which Mr Teixeira announced his "permanent resignation".
Mounting speculation
Mr Teixeira said he was leaving with a sense of "mission accomplished".
"Football in our country is associated with two things: talent and disorganisation," he wrote.
He said he had done what was within his reach, sacrificing his health.
Last week, Mr Teixeira announced he was taking leave of absence for medical reasons. However, he did not specify the nature of his health problems.
Mr Marin said nothing would immediately change at the football federation or at the World Cup organising committee under his command.
Pressure started growing on Mr Teixeira to resign after new corruption allegations surfaced earlier this month.
But state federations voted unanimously to confirm him in the posts.
In 2001, the Brazilian Congress investigated Mr Teixeira on suspicion of 13 crimes, including tax evasion, money laundering and misleading lawmakers, but no charges were brought.
He has also been investigated for allegedly taking kickbacks from a marketing firm that worked closely with Fifa in the 1990s.
And the English Football Association accused him of improper conduct during bidding for the 2018 World Cup.
He has denied all the allegations and has never been convicted.
The head of Brazil's football federation, Ricardo Teixeira, has stepped down from his post, days after taking sick leave.