He also said Dominic Cummings' infamous lockdown trip was a “mistake” that “clearly undermined” the government’s approach.
Lord Sedwill, who left his job last month, told the BBC: “Although we had exercised and prepared for pandemic threats, we didn't have in place the exact measures, and we hadn't rehearsed the exact measures for the challenge Covid-19 presented.
“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.”
However, Lord Sedwill, who contracted the virus himself, said he was “really proud of a great deal that we did”, including setting up the Nightingale Hospitals.
In May, Mr Johnson’s chief adviser Mr Cummings drove to County Durham after his wife developed Covid-19 symptoms.
Lord Sedwill said the saga was “clearly a difficult moment for the government" and added: “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.”
Lord Sedwill also uniquely held the role of National Security Adviser while also serving as Cabinet Secretary - which meant he worked closely with the security and intelligence agencies.
The senior civil servant stepped down amid reports of tension with members of the Prime Minister’s team.
He has been replaced by Prince William’s former private secretary Simon Case as part of a shake-up by the Prime Minister and Mr Cummings.
During the interview, Lord Sedwill also spoke about Donald Trump and said that while US politics is “pretty volatile” the UK-US relationship is “stable”.