In the Daily Telegraph, Lord Hague said he was asked the question "for months... everywhere I went abroad" if "we would lose heart" about leaving.
He said he explained to them that "this really is a democracy".
Lord Hague also backed a "transitional" withdrawal from the EU saying it had "immense" attractions.
Lord Hague - who campaigned to remain in the EU - stood down as foreign secretary in 2014, and left the House of Commons in 2015.
He wrote in Tuesday's Telegraph: "The electorate voted to leave the EU, and therefore we leave.
"What is more, the number of people who voted to do so was higher than the number of votes cast for any government in our history.
"To me and many of my former colleagues in government who preferred to remain, the argument was over.
"In the recent general election, both main parties were clear that they were committed to the referendum outcome.
"Globally, the message has now got through."
But Lord Hague added that "just as the message was accepted, the voters pulled off another surprise and refused to give a majority to the ministers negotiating the exit" [in the general election].
He said there was the clear potential for Brexit to become the "greatest economic, diplomatic and constitutional muddle in the modern history of the UK, with unknowable consequences for the country, the government and the Brexit project itself".
And he said the Chancellor Philip Hammond deserved "great credit" for putting forward a possible solution.
Lord Hague said: "He has evidently been trying to persuade his cabinet colleagues that we should be seeking to stay in the EU single market and customs union during a transition and 'implementation' phase lasting to 2022, followed by a free trade deal with our former partners after that.
"This is seen by longstanding advocates of leaving as a 'soft' position or a climbdown.
"But in reality it is a plan to rescue Brexit from an approaching disaster."
Mr Hammond has said any transitional deal in the period after Brexit must end by June 2022, the time of the next general election.
But the chancellor said there must be "business as usual, life as normal" for Britons as the UK left the EU.
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Business leaders and politicians asked former Foreign Secretary William Hague how the UK would "get round" the EU referendum result, he has revealed.