Under cross examination by prosecutor Richard Whittam QC, Mr Adebolajo replied "yes" when he was asked if he had killed Fusilier Rigby, describing it as a "military operation".
Fusilier Rigby's widow Rebecca left the courtroom in tears as Mr Adebolajo claimed the soldier was still moving after being hit by the car.
He refused to answer questions about how long he had planned the attack, but said he had not planned to run down Fusilier Rigby, saying "it just so happened that Allah caused him to cross in front of my car."
He described attacking Fusilier Rigby with a meat cleaver in graphic detail, and said the soldier was already dead by the time Mr Adebowale had joined him in attacking the soldier.
Earlier the court heard Mr Adebolajo, from Romford, east London, give his name as Mujaahid Abu Hamza and confirm he was married with six children, including a seven-year-old boy and a baby who was four days old on the day of the incident.
He said he had been brought up as a Christian by his family and his parents had taken him to church every Sunday.
"The memory that sticks in my mind... is probably every New Year's Eve in the evening around 11 o'clock we would gather around in candlelight and read passages from the Bible," he said.
He converted to Islam in his first year as a student at Greenwich University.
He told the court: "My religion is everything.
"When I came to Islam I realised that... real success is not just what you can acquire, but really is if you make it to paradise, because then you can relax," he added.