The explosion, which was initially reported as a gas explosion, injured at least 50 people in Kerch, a city of 147.000.
The National Anti-Terrorism Committee's spokesman Andrei Przhezdomsky said the building is now being combed for other explosive devices.
Numerous victims from Wednesday's attacks were teenage students who suffered shrapnel and bullet wounds.
Sergei Aksenov, the Russia-backed leader of Crimea, told Tass that reports circulating on social media of a shooting after the explosion were untrue, and said the situation on the scene was calm.
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COLOGNE, GERMANY - OCTOBER 15: Police stand outside Cologne main railway station on October 15, 2018 in Cologne, Germany. Reports speak of at least one explosion during the incident. "The suspect is under control", Cologne police said.
Law enforcement officials said they were treating the incident as a terror attack.
Kerch is linked to the Russian mainland by a bridge opened by Putin in May this year. He told Russian state television that a single suspect, a 22-year-old male student of the college, killed himself after the alleged attack.
Russian Defence Minister Sergei Shoigu said four military planes were ready to evacuate casualties for medical treatment and that space in military hospitals was available for the victims.
"They blew up the corridor, they ran around, throwing explosives, ran around with automatic rifles, opening classes and killing anybody they could find", she said sobbing.
The Kremlin's spokesman, Dmitry Peskov, told reporters that President Vladimir Putin has expressed his condolences to the victims and ordered emergency services to provide assistance. Someone broke in 30 minutes after I left the building.
Eyewitnesses said ambulances and other emergency vehicles rushed to the site after the blast, which some reports said took place in a cafeteria.