The blast at the Kabul office of Afghan Voice occurred during a morning panel discussion at the centre, with many students attending.
Sayed Abbas Hussaini, a journalist at the agency, said there appeared to have been more than one explosion during the attack, following an initial blast at the entrance to the compound.
He said one reporter at the agency had been killed and one injured.
Photographs showed what appeared to be serious damage at the site, in a heavily Shi’ite Muslim area in the west of the capital, and a number of dead and wounded on the ground.
Deputy Health Minister Feda Mohammad Paikan said 35 bodies had been brought into the nearby Istiqlal hospital.
Television pictures showed many of the injured suffered serious burns.
President Ashraf Ghani called the attack an “unpardonable” crime against humanity and pledged to destroy terrorist groups.
He said: “The terrorists have killed our people.
“The terrorists have attacked our mosques, our holy places and now our cultural centre.”
Afghan Voice has Shi’ite links but there was no immediate claim of responsibility.
Taliban spokesman Zabihullah Mujahid issued a statement on Twitter denying involvement.
The attack, the latest in a series to hit Afghan media groups in recent years, follows an attack on a private television station in Kabul last month.
Biraj Patnaik, from Amnesty International, said: ”This gruesome attack underscores the dangers faced by Afghan civilians.
”In one of the deadliest years on record, journalists and other civilians continue to be ruthlessly targeted by armed groups.”
Backed by the heaviest US air strikes since the height of the international combat mission in Afghanistan, Afghan forces have forced the Taliban back in many areas and prevented any major urban centre from falling into the hands of insurgents.