Paying tribute to his "iron will", Gibb's medical team said the singer had defied the odds to regain consciousness more than a week after he went into a coma.
However, the outlook for the 62-year-old musician remains uncertain as he receives more treatment for advanced bowel cancer, pneumonia and liver failure.
"Only three days ago, I warned Robin's wife, Dwina, son, Robin John, and brother, Barry, that I feared the worst," Dr Andrew Thillainayagam, of Imperial College Healthcare, said in a statement released by Gibb's spokesman.
"We felt it was very likely that Robin would succumb to what seemed to be insurmountable obstacles to any form of meaningful recovery. As a team, we were all concerned that we might be approaching the realms of futility.
"It is testament to Robin's extraordinary courage, iron will and deep reserves of physical strength that he has overcome quite incredible odds to get where he is now."
While Gibb remains weak and needs an oxygen mask to help him breath, doctors hope that if his condition improves they might be able to move him out of the intensive care unit.
His family has been at his bedside every day, talking and playing him his favorite music, Thillainayagam added.
Gibb founded the Bee Gees with his twin brother Maurice and older brother Barry in the late 1950s and went on to sell an estimated 200 million records.
The group produced some of the most memorable hits of the 1970s, including "Night Fever", "Tragedy" and "How Deep Is Your Love".