The 17-year-old, who cannot be named due to his age, stabbed James Attfield, 33, more than 100 times in Colchester on 29 March 2014.
Mr Attfield died in hospital after being found in a park in the town.
Three months later, the boy stabbed 31-year-old Saudi student Nahid Almanea 16 times as she walked towards the University of Essex.
During a hearing at the Old Bailey, the boy denied two counts of murder and possessing an offensive weapon but admitted two alternative counts of manslaughter by reason of diminished responsibility.
Det Ch Insp Morgan Cronin said he hoped the guilty plea would give the families of Mr Attfield and Miss Almanea some "comfort by finally establishing who carried out their atrocious killings".
"Detectives worked tirelessly to piece together the evidence of how they died and to establish whether there was any connection between their deaths," he said.
"Sadly we now know that one person, a 17-year-old boy, was indeed responsible."
Mr Attfield was found fighting for his life near Lower Castle Park, and later died in hospital.
He had suffered a severe brain injury when he was hit by a car in 2010, leaving one side of his body weak and affecting his reasoning and speech.
His mother Julie Finch said he was "shy and polite".
Miss Almanea was attacked on Salary Brook Trail near the University of Essex, where she was studying English.
In a statement, her family described her as a "remarkable and gentle person who was loved for her kind and caring nature".
By Gareth George, BBC Look East
After the first killing, many in Colchester were nervous about walking home alone at night, particularly when many streetlights were switched off at midnight.
Anxiety deepened after the second stabbing. Nahid Almanea was a student at the University of Essex, and the footpath where she was found is used each day by hundreds of students.
The fact both victims were stabbed so many times heightened fears. Repeated appeals by Essex Police suggested detectives were struggling to find a breakthrough, and there was disbelief when someone so young was eventually arrested.
A teenager has admitted killing two people in separate attacks while they were out walking in a town in Essex.