The two children were rescued after they apparently managed to free themselves from the vehicle as it entered the water.
Both are understood to be uninjured but were deeply shocked by their ordeal.
A search is underway for a third person, reported to a female, who is suspected to have been trapped in the car after it entered the river.
The incident occurred shortly after 7pm at Kennedy Quay just off Cork city centre.
Gardaí, Cork Fire Brigade and Cork River Rescue units were at the scene within minutes.
The Coast Guard's Rescue 117 also responded to the alert and provided air support for the major unfolding search operation.
A number of pedestrians who were in the area also raced to help.
The two children - understood to be aged 11 and 10 years - were given precautionary medical treatment at the scene.
They were later taken to Cork University Hospital (CUH) for a full medical assessment.
It is expected they will be reunited with a family member later tonight (Friday) before being allowed return to their Cork city home to recover from their ordeal.
Minutes after entering the water, the vehicle sank.
Onlookers reported that a female was trapped inside the vehicle as it vanished from sight.
The incident occurred shortly before high tide along the city quays.
Initial reports indicated that the woman and two children had been returning home after visiting a city restaurant when the accident occurred.
Divers are expected to begin operations at the scene and will attempt to examine the vehicle on the river bed to determine if the missing person is still trapped inside.
It is hoped they will then be able to attach cable lines to the car to allow it to be pulled from the river.
The area has been cordoned off to facilitate the work of the emergency services.
Searches are also being conducted for some distance downstream from Kennedy Quay given the strong tidal currents involved.
A Garda investigation into the full circumstances of the incident is now underway.
CCTV security camera footage from around the area will be examined by gardaí to determine the precise movements of the vehicle involved in the minutes leading up to the accident.