The bodies of 11 people who died at the facility after being caught in crossfire or failing to receive medication were also retrieved, the humanitarian organisation said.
Syria's army has recaptured around 80 percent of the former rebel bastion of east Aleppo since beginning an assault three weeks ago, and tens of thousands of civilians have been displaced.
The International Committee of the Red Cross said it carried out an operation overnight with Syria's Red Crescent (SARC) to evacuate 150 patients and civilians from the Dar al-Safaa facility in the Old City.
Around three dozen other civilians, some of them injured, had also sought refuge there, it added.
"These patients and civilians had been trapped in the area for days because of heavy clashes nearby and as the front line kept drawing closer," said ICRC Syria delegation head Marianne Gasser.
"Many of them cannot move and need special attention and care. It must have been terrifying for them," she added.
The two organisations had been trying to evacuate the facility since Tuesday, but were forced to postpone initial efforts because of heavy fighting, the statement said.
"They were either caught in the crossfire or died because they did not have access to the right medications."
Syria's government is three weeks into an operation to recapture east Aleppo - in rebel hands since 2012.
It now holds around 80 per cent of the former opposition stronghold, and has advanced quickly.
At least 384 civilians, including 45 children, have been killed in government fire on east Aleppo since the operation began, according to the Syrian Observatory for Human Rights.
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