Mr Swinney said the case of Richard Brown, 55, who died alone in the stairwell of his tenement building after repeatedly calling an ambulance, was a “serious issue”.
Speaking at First Minister’s Questions on Thursday while filling in for Nicola Sturgeon, Mr Swinney said an investigation into his death had been launched.
A survey from union Unite published on Thursday found more than half of ambulance staff have witnessed “adverse clinical events” – patients dying or becoming seriously ill – due to lengthy waiting times.
Meanwhile, one in six said patients have had to wait 20 hours or more from a 999 call to treatment on their callouts.
Scottish Conservative leader Douglas Ross was the first to raise the case of Mr Brown, asking Mr Swinney: “Deputy First Minister, why is this happening in Scotland today?”.
He called for the SNP to set out exactly how much funding would be delivered to the NHS as part of the Budget.
Mr Swinney replied: "First of all, I want to express my sympathy to Mr Brown's family, because Mr Brown should not have had the experience that he had. And I'm very sorry to Mr Brown's family and what they are enduring in addition to the loss of Mr Brown.”
However, he defended the Scottish Government’s actions when questioned by opposition party leaders on a mounting crisis in the NHS, accusing Scottish Labour leader Anas Sarwar of failing to “recognise the reality” of the pressures faced by the health service due to Covid-19.
Liberal Democrat leader Alex Cole-Hamilton raised information from a Freedom of Information request, which showed the number of ambulance staff hours lost to mental ill health is up 300 per cent since 2017.
He claimed there was a “muscle memory” to opposition leaders’ exchanges with the government over the crisis in the NHS.
Mr Cole-Hamilton said: “Week after week opposition members ask the government about the crisis in emergency care and week after week the government respond by blaming the pandemic.
“The Deputy First Minister has doubled down on that today by accusing Anas Sarwar of being in denial, and yet the former chief executive of NHS Scotland said this crisis has been years in the making, but the pandemic has only hastened the date.”
He added: “I want the Deputy First Minister to put himself in the shoes of our hard-working emergency care staff, the call handlers answering repeated calls asking again and again when the ambulance will come.”
Mr Sarwar asked Mr Swinney to commit to a maximum 30-minute turnaround for ambulances outside hospitals and claimed Covid had only exacerbated the problems in the NHS.
"Listen to NHS staff, don’t ignore them," he said. "Covid can't be the cover. Let's not pretend these problems started with Covid."
Mr Swinney said the government had put in place a £300 million NHS and winter care package across the health service back in September, such as recruiting 1,000 NHS staff and boosting pay of social care staff.