Christopher Groves, 38, and his son Connor, nine, died in the accident on the A725 near the Raith interchange, South Lanarkshire, at around 11.15am on Saturday.
They lived in Blantyre, North Lanarkshire.
Connor, a pupil at Udston Primary School in Hamilton, was described by his head teacher, Elaine Henry, as a "bright, friendly, happy boy who was valued by all in the school community".
She said: "The school is shocked and saddened to learn of the tragic loss of Connor and his father.
"The school was privileged to have had the opportunity to be part of Connor's life. He was indeed a credit to his family, and our thoughts and prayers are with his mum, family and friends."
Mr Groves and Connor were travelling south from Whistleberry Toll towards East Kilbride when their blue Honda S2000 left the road and went into the river.
Police divers removed Mr Groves and his son from the car but they were pronounced dead at the scene.
A special service will be held at Udston Primary today to remember Connor, who was an only child.
The father and son were formally identified yesterday by Connor's mother.
Mourners, including Connor's classmates, packed out Hillhouse Parish Church in Hamilton yesterday as the community came to terms with the tragedy.
Reverend David Burt, who is also chaplain of Udston Primary, said: "We had virtually a whole turnout of staff, and not just teachers - dinner ladies, cleaners, everyone had come along.
"The mood was very sombre - a lot of people in shock.
"I knew a lot of his friends because they attended our Boys' Brigade and Sunday School.
"They were in tears. One of the parents said to me 'how do I tell my nine-year-old that his best friend is dead?."
Investigators were on the scene until late on Saturday, but it is unclear what caused the crash, although it is believed no other cars were involved.
Police are now appealing for witnesses to contact them on 101.