It was not clear how many children would be affected.
The decision was an abrupt turnaround months before Corbett will ask voters to give him a second term.
A career prosecutor, Corbett had long rejected any legalization of medical marijuana. A spokeswoman on Thursday said he remained opposed to the broader use of the drug to treat other diseases or pain-related disorders. In his statement, Corbett said he had a responsibility to protect the health and safety of all Pennsylvanians.
Still, advocates were cautiously optimistic.
State Sen. Mike Folmer (R., Lebanon), a cosponsor of legislation to legalize medical marijuana, praised th e governor's decision, and called it "a baby step" on the road to helping sick people who could benefit from drugs derived from the marijuana plant.
"The devil is in details about what this means," Folmer said. "It does not go far enough for all sick people - soldiers with PTSD, people with diabetes, cancer, glaucoma."
His cosponsor on the bill, State Sen. Daylin Leach (D., Montgomery), on Monday had threatened a sit-in outside Corbett's Capitol office with families of children with seizure disorders.
Leach said the unresolved details included when the program would start, what the criteria would be for acceptance, and whether it would be open to all families with children with seizure disorders.
"Families want to know specifics," he said.
Spokeswoman Christine Cronkright said administration officials were already speaking with officials at Children's Hospital of Philadelphia, which is conducting a pilot study on the use of Epidiolex, a form of CBD, and had reached out to legislators.
Senate Majority Leader Dominic Pileggi (R., Delaware) also commended the governor's action.
"If a child's physician believes that cannabidiol would relieve suffering, state law should not stand in the way," Pileggi said. "I will continue to work with Sen. Folmer on his legislation and look forward to sending a bill to the governor's desk."
Christine and Eric Brann, Dauphin County residents who were among the parents to meet Thursday with Corbett, are pushing for access to a medication made from the whole marijuana plant.
"As parents, we explained that CBD is not going to be sufficient," said Christine Brann, whose 3-year-old son, Garrett, has Dravet syndrome, a form of epilepsy.
The boy is on a waiting list for the medication derived from the whole plant in Colorado, but she would have to move there to get it for him.
Brann said she was disappointed that Corbett had staked out a middle ground but "grateful that he opened the door to this dialogue."
At the American Academy of Neurology's annual meeting this week in Philadelphia, researchers released a review of studies on the effectiveness of medical marijuana for neurological conditions, including epilepsy.
The review found that there was not enough evidence to say whether medical marijuana is effective for treating epileptic seizures. Barbara S. Koppel, a neurologist at New York Medical College, who wrote the review, said there were only two usable papers on epilepsy.
Studies in animal models suggest that cannabidiol acts as an anticonvulsant.
The Pennsylvania State Nurses Association, representing more than 212,000 registered nurses, applauded Corbett "for listening to patients, parents, and professional nurses on the compassionate use of medical cannabis."
Political analyst and pollster G. Terry Madonna of Franklin and Marshall College said Corbett was likely driven to act by two forces: compelling stories about sick children, and politics.
"It's impossible to listen to the gripping emotional testimony and not be moved," said Madonna, who this year released a poll that found more than 80 percent of Pennsylvania voters supported legalization of medical marijuana. "Politically, it's the right thing to do."
The governor's decision is among several in the last eight months that reflect a move to the center.
In September, Corbett announced he would support taking federal Medicaid money to help low-income Pennsylvanians get health insurance through a not-yet-approved, state-specific plan to use federal money to pay private insurers.
In December, he said he would support legislation banning discrimination against gay people in employment and housing.
Inquirer staff writer Stacey Burling contributed to this article.