Images posted on social media showed flames emerging from many of the building's 24 floors.
There were no immediate reports of fatalities from the blaze, the cause of which is unknown.
A number of people were treated for smoke inhalation at the scene, officials told the Gulf News newspaper.
Residents were evacuated from the building and the power was cut, Brigadier Abdullah Al Suwaidi, head of civil defence in Sharjah, told Gulf News.
Around 200 families live in the tower, the newspaper said.
Images suggested that the fire started on one of the building's lower floors before spreading rapidly upwards. Flaming debris could be seen falling into the adjacent car park.
The blaze reportedly brought traffic to a near standstill in the roads around the building.
A high-rise residential building in Sharjah in the United Arab Emirates has caught fire, sending plumes of thick black smoke into the air.