Burnet, 28, secured silver in the mixed Nacra 17 class at the Tokyo Olympics alongside her sailing partner John Gimson, 38, on Tuesday morning.
Louise Burnet, from Shandon, Gare Loch, in Argyll and Bute, said her daughter has “always had this great passion” since she began sailing at the age of five in an Optimist dinghy.
Mrs Burnet, 62, said “we would never ever have dreamed of this happening”, adding: “We’re completely over the moon, it’s an awesome result.”
The retired musician said: “They’ve just been a great team together and it’s a massive passion for them both.
“They are really good friends and you need that partner you click with.
“When Anna started sailing at the local yacht club she had no fear of the water from an early age.
“There were a lot of long drives at weekends which her father Colin did a lot of, 11 hours down to the south coast at weekends, and it is certainly all worth it now.”
The achievement came on a stellar morning for Team GB sailors, with Giles Scott defending his Olympic title to be crowned champion in the men’s Finn class and Stuart Bithell and Dylan Fletcher securing gold in the 49er class.
Burnet, who now lives in Weymouth, Dorset, started sailing at the Royal Northern and Clyde Yacht Club on Gare Loch, Scotland’s oldest yacht club founded in 1824.
She soon won the female national Optimist title and was selected for the British worlds team in 2006 at the age of 14, said the British Sailing Team.
Burnet and Gimson guaranteed a medal going into the final race and finished safely in fifth to stay in second place behind Italians Ruggero Tita and Caterina Banti.
Chief executive of sportscotland, Stewart Harris, said: “Scotland has a proud tradition of successful sailors on Team GB and it’s terrific to see Anna Burnet join that illustrious list.
“To be selected to compete at an Olympic Games is a huge achievement, but to win a silver medal is very special.
“Congratulations to Anna, her partner John Gimson and the whole Scottish and British sailing family.”