Most of the 37 survivors reached an island in the lake while others were found in a village, some of them washed ashore.
The children were on holiday from Moscow, and Mayor Sergei Sobyanin has sent doctors and psychologists.
Four people have been arrested on suspicion of safety violations.
"A great tragedy has occurred in Karelia," the mayor tweeted.
"My  condolences to their families and friends."
Rescue services say they received an alert at 11:15 (08:15 GMT) on Sunday that a tour group of 47 children and four adults was missing on Lake Syamozero after a storm.
Emergency crews using boats and helicopters were sent to the scene.
On Sunday evening, investigators announced that all of the victims were children, denying earlier reports that one adult was among the dead.
It appears that the boating party, mostly adolescents aged between 12 and 15, were staying in a summer camp near the lake, a popular outdoor tourism destination.
They had set out in boats on Saturday when the storm rose. Between 49 and 51 people were in the party.
One of the survivors, a girl aged 12, raised the alarm after reaching a local village on Sunday, a police source told Russian tabloid Life News (in Russian).
Yulia is quoted as saying the group were using two boats and a raft when the storm began, and the craft were scattered across the lake.
Washed ashore, she only recovered consciousness on Sunday, after which she walked along the shore to the village of Kudama.
Along the way, she met a boy survivor who was covered in blood and could not move.
After taking note of the spot where she had found him, she reached the village, where a local resident phoned emergency services.
A regional MP, Alexei Gavrilov, reportedly told state TV channel Rossiya 24 there had been repeated warnings in the days before the storm, advising against any boating on the lake.
"They didn't have the right to go out boating," he was quoted as saying by the Associated Press news agency.
Fourteen people, mostly children, died when a storm caught a boating party on Lake Syamozero in northern Russia's Karelia region.