But he was not asked to enter a plea to a charge of murder when he appeared at London's Old Bailey court, as medical reports were awaited.
Ms Everard, a 33-year-old marketing executive, went missing in Clapham, south London, on 3 March.
Her disappearance led to vigils and protests and prompted the government to promise enhanced police patrols at night, as well as funding to make the streets safer for women.
Couzens was wearing khaki trousers and a grey sweatshirt as he appeared by video link from Belmarsh high-security prison in southwest London, where he is being held on remand.
He responded "guilty, sir" to the charge of kidnapping Ms Everard "unlawfully and by force or fraud" on 3 March and also pleaded guilty to a second charge of rape between 2 and 10 March.
Ms Everard's family sat in the court as Couzens entered his pleas.
Ms Everard had been visiting friends in the Clapham area and was returning to her home in nearby Brixton when she disappeared.
Her body was discovered a week later in woods some 80 kilometres (50 miles) away in Kent, southeast England.
The Metropolitan Police said last week that a post-mortem examination gave the cause of death as "compression of the neck".
The next hearing is due to take place on 9 July. A provisional trial date has been set for October.
If you or someone you know is impacted by family and domestic violence or sexual assault, call1800RESPECT on 1800 737 732 or visit 1800RESPECT.org.au. The Men’s Referral Service provides advice for men on domestic violence and can be contacted on 1300 766 491. In an emergency, call 000.