Speaking on BBC Radio 4's Today programme, Mr Grayling said the document was not a government memo and rejected its contents.
"My own experience is very different to that," he said, describing the Brexit project as a "team effort".
Mr Grayling said negotiations would be "complex but by no means the challenge that is set out in today's newspaper story".
He rejected the memo's estimate of an extra 30,000 civil servants, saying: "I do not know what 30,000 people would do in this process."
Prime Minister Theresa May hopes to invoke Article 50 - beginning the formal two-year process for leaving the EU - by the end of March next year.
However, BBC political correspondent Chris Mason - who has seen the memo - says the document shows how "complex, fraught and challenging delivering Brexit will be".
The leaked memo - written by an un-named consultant and entitled "Brexit Update" of 7 November - suggests it will take another six months before the government decides precisely what it wants to achieve from Brexit or agrees on its priorities.
The report criticises Mrs May, who it says is "acquiring a reputation of drawing in decisions and details to settle matters herself" - an approach it describes as being "unlikely to be sustainable".
The document also identifies cabinet splits between Foreign Secretary Boris Johnson, Brexit Secretary David Davis and International Trade Secretary Liam Fox on one side, and Chancellor Philip Hammond and Business Secretary Greg Clark on the other.
It says: "Every department has developed a 'bottom-up' plan of what the impact of Brexit could be - and its plan to cope with the 'worst case'.
"Although necessary, this falls considerably short of having a 'government plan for Brexit' because it has no prioritisation and no link to the overall negotiation strategy."
The memo also suggests the government does not have enough officials to implement Brexit quickly, while departments are developing individual plans resulting in "well over 500 projects".
It estimates an additional 30,000 extra civil servants could be required to meet the workload.
The document also says big businesses could soon "point a gun at the government's head" to secure what they need to maintain jobs and investment.
It comes after Japanese car manufacturer Nissan said it had been given "support and assurances" over trading conditions once Britain leaves the EU.