More than 100 firefighters were deployed to battle the blaze, which began at the complex in Titisee-Neustadt early in the afternoon, said Markus Straub, a spokesman for local firefighters.
Ambulances races to the scene as smoke poured from the windows of the modern, three-story center in one of the country's deadliest blazes in recent memory. German Chancellor Angela Merkel is "shocked" about the loss of so many lives, her spokesman said on Twitter.
The cause of the fire was not immediately known, police and firefighters said.
The center usually has around 120 people at any one time, said Mirko Steffl, a police spokesman in the nearby city of Freiburg.
It was not immediately clear how many people had to be rescued, but by late afternoon workers had determined that no one else remained trapped in the building, he said.
The center employs people with mental or physical disabilities in a variety of jobs, including metalwork, woodwork and electrical installation, and it is run by the Catholic Church's Caritas organization.
"The biggest problem this afternoon was the smoke situation," said Alfred Oschwald, another Freiburg police spokesman. Smoke inhalation is believed to be the likely cause of the deaths of the 13 disabled people and one of the facility's social workers, he told broadcaster n-tv. Eight people were hurt, but none of them had life-threatening injuries, he said.
Some 25 psychological counselors were assisting those rescued from the building and the victims' relatives in a nearby warehouse, Steffl said.
"It will take days to investigate what caused the fire," police spokesman Karl-Heinz Schmid said.
"The fire spread unusually fast," Baden-Wuerttemberg state's fire chief Hermann Schroeder told public broadcaster ARD. Only about two minutes after the automatic fire alert was triggered, neighbors started calling and reported strong smoke coming from the building, he said.
Armin Hinterseh, the mayor of Titisee-Neustadt, a popular lakeside tourist destination amid the rolling hills of the Black Forest in southwestern German Baden-Wuerttemberg state, said the center's buildings are quite new. "It is devastating. We now have to find out how it happened," Hinterseh was quoted as saying by the local daily Badische Zeitung.