The strike on the Al-Marja neighbourhood wounded several people, including at least one nurse, the civil defence, known as the White Helmets, said.
The clinic had been providing dental services and treatment for chronic illnesses for about five years.
Earlier this week, 30 people were killed -- including two doctors -- when air strikes hit the Al-Quds hospital and a nearby block of flats in the rebel-held Sukkari neighbourhood.
The attack on the hospital, which is supported by Doctors Without Borders (MSF), has been roundly condemned by world leaders, including US Secretary of State John Kerry who said it appeared to be a "deliberate strike."
More than 200 civilians have been killed in Aleppo over the past week as rebels have pounded government-held neighbourhoods with rocket and artillery fire and the regime has hit rebel areas with air raids.
On Friday, air strikes on rebel-held districts killed two civilians, one of them a child, the Syrian Observatory for Human Rights said.
MSF released a statement regarding the attack on Wednesday:
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The bombing of the MSF-supported Al Quds hospital in the northern Syrian city of Aleppo on Wednesday night has killed at least 14 people, including at least two doctors.
According to hospital staff on the ground, the hospital was destroyed by at least one airstrike which directly hit the building, reducing it to rubble. Other airstrikes in the neighbourhood also hit areas close to the hospital.
“MSF categorically condemns this outrageous targeting of yet another medical facility in Syria” said Muskilda Zancada, MSF head of mission, Syria. “This devastating attack has destroyed a vital hospital in Aleppo, and the main referral centre for paediatric care in the area. Where is the outrage among those with the power and obligation to stop this carnage?”
The situation in Aleppo city, consistently at the frontlines of the brutal conflict, was critical even before this attack. An estimated 250,000 people remain in the city, which has seen dramatic increases in levels of bombardments, fighting and fatalities in recent weeks. Only one road remains open in and out of the non- government held areas. If it gets cut off, the city will become besieged.
Over the last week, several other medical structures have been attacked and destroyed in Aleppo, and five rescue workers from the Syrian Civil Defence organisation have been killed.
MSF has been donating medical supplies to Al Quds hospital since 2012, and has built a very strong working relationship with the staff there.
“Compounding this tragedy is that the dedication and commitment of the staff of Al Quds, working under unimaginable conditions, has been unwavering throughout this bloody conflict,” said Zancada.
The 34-bed hospital offered services including an emergency room, obstetric care, an outpatients department, an inpatients department, an intensive care unit and an operating theatre. Eight doctors and 28 nurses worked full time in the hospital, which was the main referral centre for paediatrics in Aleppo.