Seven people were killed and 51 injured when a tram derailed as it entered a sharp bend at almost four times the speed limit on November 9 last year.
A number of survivors and families of the victims are suing TfL, which manages the network, and Tram Operations Limited (TOL), a subsidiary of First Group, which is responsible for the day-to-day operation of the trams.
Insurers for TfL and TOL wrote to a law firm handling a damages claim in relation to Mark Smith, 35, who was among the victims.
They advised the law firm that the letter was an "admission of liability for the purposes of your client's civil claim".
The insurers said they were "not in a position to comment" on specific allegations of negligence, and added that "all interim payments requested" have been made to the claimant.
Richard Geraghty, a specialist serious injury lawyer from Slater and Gordon, which represents two of the victims, said: "Our clients are relieved that the defendants have admitted liability in the Croydon tram crash case.
"The trauma they have been through as a result of the crash has been difficult for them to come to terms with and the news that they will not have to endure a civil trial is very welcome.
"As there is a criminal investigation ongoing it would be inappropriate for us to comment further, but our clients are anxious to find out the full facts of what happened and what caused the crash that devastated their lives."
An interim report by the Rail Accident Investigation Branch said the tram was travelling at 46mph as it entered the bend, which had a 13mph limit.
The driver, Alfred Dorris, 42, from Beckenham, south-east London, was arrested at the scene and questioned on suspicion of manslaughter before being bailed until May.
Jonathan Fox, TfL's director of London rail, said: "The cause of the tragic derailment at Sandilands last November is not yet known and we continue to assist with the ongoing investigations.
"This is clearly a terribly difficult time for everyone affected.
"We have been in touch with everyone injured who has notified us of a claim and with the dependants of the people who lost their lives to confirm that liability is admitted in respect of their civil claims.
"We urge anyone needing further help to contact us straight away."