The west London centre will start in the next couple of weeks and be aimed at people with gaming disorders.
Other internet-based addictions could be treated in future, however.
Psychiatrist and clinic founder Henrietta Bowden-Jones said the centre will seek to provide treatment, do research and give advice to families.
She told The Guardian: “Gaming disorder is finally getting the attention it deserves.
“The distress and harm it can cause is extreme and I feel a moral duty on behalf of the NHS to provide the evidence-based treatment these young people and their families need.
“We are unlikely to witness an epidemic of young players with an addiction to gaming but for the ones who do struggle, the centre for internet disorders will be a life-changer.”
A spokesman for the Central and North West London NHS Foundation Trust could not confirm the numbers of patients who are being signed up for treatment.
The news comes after the World Health Organisation classified gaming disorder as an addiction, describing it as a pattern of persistent gaming behaviour so severe it “takes precedence over other life interests”.
Symptoms include impaired control over gaming and increased priority given to gaming.
WHO said: “Studies suggest that gaming disorder affects only a small proportion of people who engage in digital or video-gaming activities.
“However, people who partake in gaming should be alert to the amount of time they spend on gaming activities, particularly when it is to the exclusion of other daily activities, as well as to any changes in their physical or psychological health and social functioning that could be attributed to their pattern of gaming behaviour.”
But experts criticised the move for making a hobby seem abnormal.