The blast took place in the city of Kerch.
Law enforcement officials said they were treating the incident, in which numerous victims were teenage pupils, as a terrorist attack.
TASS, also state-run, quoted an unnamed law enforcement official as saying that about 70 people were injured and an unknown number of them had died.
Initially, emergency officials said the blast was believed to have been the result of a gas canister explosion.
The attack had first been reported as an explosion, with Russia's National Anti-Terrorism Committee saying an "unidentified explosive device" had gone off.
Committee spokesman Andrei Przhezdomsky said sappers were inspecting the building for other explosive devices.More news: Caroline Garcia beats Pliskova to win Tianjin for first title of 2018
Russia's Investigative Committee, the nation's top investigative agency, said the device that went off at the college's canteen was rigged with shrapnel.
Speaking to local media, the head of the college Olga Grebennikova said someone blew themselves up inside the college, leaving "the bodies of children everywhere".
"Then a strong explosion went off, but thank God, I was already outside and saw our guys being thrown out of the windows by the explosive wave", he said, adding that part of the building collapsed. They also beefed up security at a new 19-kilometer (11.8-mile) bridge that links the peninsula with Russian Federation that opened earlier this year.
President Vladimir Putin ordered the Federal Security Service to investigate the explosion, his spokesman, Dmitry Peskov, said.
Military units were deployed around the college.
It said in a statement that 13 people had been killed and around 50 wounded, many of whom were teenagers.
Russia's Health Minister Veronika Skvortsova headed to the area to help coordinate assistance to the wounded and helicopters carrying emergency medical teams flew to the area.