Staff were handling 40 calls a minute during peak periods on Christmas Day, Boxing Day and Bank Holiday Monday.
More than 14,300 calls were received on Monday, making it NHS 24's busiest day since January 2007.
The surge in demand for the service was due to the closure of GP surgeries over the festive weekend.
NHS Scotland has urged people to be prepared to cope with winter illnesses ahead of further surgery closures over Hogmanay.
NHS 24 medical director Dr George Crooks asked the public to protect themselves by following health guidelines in the Be Ready For Winter campaign, including ensuring medicine cabinets are well stocked to cope with seasonal colds and flu-like illness.
He said: "It's important people know who to turn to if they are unwell. We're advising people to seek the pharmacist's advice as the first step, but to make an appointment with their GP if they have an illness or injury that won't go away.
"Pharmacists provide a range of services to deal with minor illnesses and ailments, and in many cases can provide something over the counter to help ease symptom.
Of course, prevention is always better than cure.
"By making sure you eat well, get lots of sleep, wash your hands regularly and dispose of tissues properly after a single use, you can help prevent illness ruining your celebrations this New Year."