The explosion occurred at 8.52 a.m. near an entrance to the FSB directorate. According to preliminary information, the suspect was a local teenager who had gone into the building and took out an "unidentified object" from his bag, which exploded in his hands, the Investigative Committee was cited as saying by Tass news agency.
"The teenager was identified," the committee said. He died of his wounds. Investigators were treating the attack as an act of terrorism.
A CCTV image of the suspect appeared on social media soon afterwards, along with a message he allegedly sent just before the blast.
Although the authenticity of the message posted on an anarchist chat group was not confirmed, the user identified himself as an anarcho-communist and claimed the FSB "fabricated cases and tortured people".
Governor Igor Orlov said local authorities in Arkhangelsk had taken measures to heighten security in all public buildings.
Attacks on security services are unusual in Russia, although the police have been targeted in parts of the Caucasus, including Chechnya and Dagestan.
(With inputs from agencies.)