The 22-year-old was driven by police from Strand Road station to the court on Bishop Street on Wednesday morning.
The vehicle was parked in a secure area outside the building, but when the man got out he climbed over a six-foot high security fence.
A police officer tried to catch him, but the man then scaled a second security fence, this one ten feet high.
He then walked through the turnstiles and sprinted along the street.
He was due to appear in court on Wednesday charged with assaulting a couple and stealing Christmas presents and money from them.
It is understood the incident was recorded by CCTV cameras outside the courthouse.
Police could not confirm whether the man was handcuffed or not, but said they would be carrying out an investigation into his escape.
A police search is underway in Londonderry after a man escaped from custody outside the city's court.