The process was expected to take six to eight hours, reports CBS News correspondent Robert Berger. The doctors then will conduct tests to see how Sharon responds.
Israel's Channel 10 reported that Sharon moved one of his hands as doctors roused him. "He coughed and moved," the Ynet Web site quotes Sharon aides as saying. "The situation is much better than before. We're happy about the good signs."
Outside experts said that while independent breathing meant Sharon had better chances for survival, it gave no indication about his other physical or mental capacities.
Doctors made the decision to lift the anesthesia after a round of consultations Monday. Hospital director Dr. Shlomo Mor-Yosef said the process of weaning Sharon from sedation could take hours or days.
"As soon as we started reducing the drugs ... the prime minister started to breathe independently, although he is still hooked up to a respirator that is used as an aid," Mor-Yosef said. He added that Sharon remains in critical condition.
Sharon suffered a severe stroke on Wednesday, two weeks after a first, mild stroke, and was rushed to Hadassah where he has undergone two surgeries to stop bleeding in his brain.
Doctors have kept him in a medically induced coma and on a respirator since Thursday to give him time to heal from the trauma of the stroke and the surgeries.
After withdrawing the sedatives, doctors were to pass their assessment of brain damage to Attorney General Meni Mazuz, who will then decide whether to declare the prime minister permanently incapacitated.
"The minute we know what damage has occurred, we will talk," Justice Ministry spokesman Yaakov Galanti said.
Since an acting prime minister is in place, there is no urgency to such a declaration, Galanti added.
Ehud Olmert, Sharon's deputy, has been named acting prime minister and can serve in that role for 100 days.
Outside experts have said doctors should have a good idea of the extent of the damage by the end of the day. One of Sharon's neurosurgeons has cautioned that it was unlikely he could function as prime minister again.
"My guess is he'll be in the ICU and on a respirator for quite some time," Dr. Keith Siller, head of neurology at New York University Medical Center, told CBS News' Tony Guida.
Because he is a head of state, it would have been inconceivable for doctors not to attempt every intervention, but Siller said that might not be best for Sharon.
"Often doing something is wrong," Siller said. "This is a bad gamble with little likelihood of success."
The 77-year-old Sharon, Israel's most popular politician, was seen by many here as the best hope for resolving the Israel-Palestinian conflict. His abrupt illness and expected departure from the Mideast political stage has raised concern that momentum on territorial concessions, created by his recent Gaza Strip withdrawal, would stop, and that Sharon's successor wouldn't have the stature to forge ahead on drawing Israel's final borders.
In the event the attorney general declares permanent incapacitation, the Cabinet would have to elect a new prime minister within 24 hours, from among the five sitting Cabinet ministers from Sharon's Kadima Party who are also lawmakers, Galanti said.
That group includes Olmert, a potential political heir.
Before his collapse, Sharon appeared headed to a landslide victory in March 28 elections at the head of the Kadima Party, which seeks further pullbacks while strengthening Israel's hold over major settlement blocs.
Olmert told the Cabinet on Sunday that he would work to carry on Sharon's political legacy.
Doctors not involved in Sharon's care said that if he awakens, the extent of his responses could vary widely, from slight movements of the fingers or opening of the eyes, to a much fuller awakening. They have also cautioned that there is no guarantee that Sharon will awaken from the anesthesia.
That Sharon can breathe on his own "tells us that one part of his brain is functioning, the respiratory center," said Dr. John Martin, a professor of cardiovascular medicine at University College in London. "It doesn't tell us how he is thinking, it doesn't tell us how he can speak, it doesn't tell us how he can move his arms and legs.
"His chances of survival are better than if the respiratory center had been damaged, but that still doesn't mean he's going to survive. ... It is still highly probable that he will die," Martin added, noting that Sharon's weight and age work against him.
One of Sharon's surgeons, Dr. Jose Cohen, said that while Sharon's chances of survival were high, his ability to think and reason would be impaired.
"He will not continue to be prime minister, but maybe he will be able to understand and to speak," Cohen said in comments published Sunday by The Jerusalem Post.
Israel's Cabinet met for its weekly gathering Sunday for the first time since Sharon's stroke.
Olmert, sitting next to Sharon's empty chair, promised to "carry out the wishes of Sharon."