Swinney visited NHS 24 and Scottish Ambulance Service staff who are preparing for the busy festive period.
He urged people to ensure they get the right care at the right place by checking the NHS inform website in the first instance if they become unwell.
People should call 111 for urgent but non-life-threatening concerns, mental health support, or advice when local services are closed.
Swinney said: “We know that demand for the NHS increases quite substantially over the festive period and there are a number of steps you can take to help yourself if you become unwell.
“A&E is expected to be exceptionally busy in the coming weeks so only attend if you have a genuine emergency that requires urgent medical treatment.”
Scottish Ambulance Service chief executive Michael Dickson said: “Winter is always an exceptionally busy period and as demand on our service increases, we’re asking the public to use our service responsibly by only phoning 999 in a life-threatening emergency.
“Currently over 50% of our calls are managed without having to take patients to A&E. However, for those who do need hospital treatment, system-wide pressures, including lengthy handover times in key areas, are having a direct impact on our ability to get crews back on the road to reach other patients.”