Christopher Groves, 38, and his son Connor, from Blantyre, died in the accident on the A725 near the Raith interchange, South Lanarkshire, at around 11.15am yesterday.
Their car was travelling south from Whistleberry Toll towards East Kilbride when it left the road and went into the river.
The car was travelling on a flyover over the Clyde when the accident occurred, and not over Bothwell Bridge as earlier reports indicated.
Police divers removed Mr Groves and his son from the car but they were pronounced dead at the scene.
A Police Scotland spokeswoman said: "A search of the area and water took place, and a short time later two people were removed from a vehicle in the water.
"Sadly, one man, aged, 38 years and a boy aged nine years were pronounced dead at the scene."
Their relatives have been told and a report will be sent to the procurator fiscal.
The road was closed for several hours while officers investigated the crash.
Police appealed for anyone who witnessed the incident to contact them on 101.