The former cabinet secretary, who was forced out by the prime minister in June, also acknowledged the government did not have the “exact measures” in place to confront Covid-19.
“I think there is a genuine question about whether we could have been better prepared in the first place and that is obviously a very legitimate challenge,” Lord Sedwill said.
It would be for a future inquiry to determine whether the lockdown in March came too late and whether ministers had prepared properly to cope with a pandemic.
However, the prime minister has refused to start that inquiry – despite promising it in July – amid suspicions he is dragging his heels to avoid possible heavy criticism.
In April, Mr Cummings left London for his second home in Durham, when both he and his wife were showing symptoms – and then claimed he drove to Barnard Castle to test his eyesight.
A survey last month found that the public strongly believed Mr Cummings’ actions had “undermined compliance with lockdown rules” – a view backed by 76 per cent of voters.
Almost three quarters (74 per cent) said the prime minister was “wrong” to stand by his de-facto chief-of-staff when his apparent rule-breaking was exposed.
Lord Sedwill, speaking to the BBC, said of the incident: “It was clearly a difficult moment for the government. It was a mistake. Whether everyone should quit every time they make a mistake, I don't think is right.
“But it clearly undermined the government's coherent narrative about people following the rules.”