Boris Johnson has told his fellow Brexiteers they should not "gloat" about the UK's departure from the EU, which he said was a cause for "hope not fear".
The foreign secretary urged people to "unite about what we all believe in", an "outward-looking, confident" UK.
Leaving the EU was not a "great V-sign from the cliffs of Dover", he said.
Mr Johnson also said the result cannot be reversed and that Britain should not be bound by EU rules after Brexit.
And he questioned the economic benefits of being in the EU single market and customs union, which the government plans to leave.
It was the first in a series of speeches by Theresa May and her ministers designed to map out "road to Brexit".
Mr Johnson was one of the leading figures in the 2016 Leave campaign, and has previously been accused of undermining Prime Minister Theresa May's Brexit strategy.
But he stuck to the government's official negotiating position during his speech in central London.
Johnson's message for Remainers
In seeking to build bridges with the other side of the EU debate, Mr Johnson said he risked "simply causing further irritation" and accepted he would not "persuade everybody" but added: "I have to try. In the end these are people's feelings and people's feelings matter."
"We must accept that many [Remainers] are actuated by entirely noble sentiments, a real sense of solidarity with our European neighbours and a desire for the UK to succeed," he said.
"If we are to carry this project through to national success - as we must - then we must also reach out to those who still have anxieties.
"I want to try to anatomise at least some of those fears and to show... that Brexit is not grounds for fear but hope."
According to Mr Johnson, Brexit is "not some great V-sign from the cliffs of Dover", but "the expression of legitimate and natural desire to self govern of the people".
"That is surely not some reactionary Farageist concept," he added in a reference to former UKIP leader Nigel Farage, who hit back on Twitter.
As for @BorisJohnson’s comment on “Faragiste reactionaries”, he should be careful - 72% of the public wanted tighter immigration controls at the time of the EU referendum. They now want to see a clear plan.— Nigel Farage (@Nigel_Farage) February 14, 2018
Echoes of the referendum campaign
Alongside his calls to Brexit supporters not to "gloat" and "sit back in silent satisfaction", Mr Johnson said holding another referendum on the UK's membership of the EU - as some campaigners are calling for - would be a "disastrous mistake that would lead to permanent and ineradicable feelings of betrayal".
He frequently used variants of the 2016 referendum's "take back control" argument - on things like regulations and tariffs so businesses did not have laws affecting them "imposed from abroad" when they have no power to elect or remove the people making them.
It would be "intolerable and undemocratic" if the UK was subject to EU laws after Brexit, he said.
Mr Johnson said the benefits of being in the single market and customs union were "nothing like as conspicuous or irrefutable" as claimed by their supporters, saying other countries were able to trade with the EU without paying membership fees.
However, during a transition period immediately after the UK leaves in March 2019 things would "remain as they are", he said.
Reaction from Remain-backers
Pro-EU campaigners hit back at his overtures to Remain voters - with Labour MP Chuka Umunna describing the speech as an "exercise in hypocrisy".
Mr Umunna, of the anti "hard Brexit" Open Britain campaign, said: "We are already a great country, we are already internationalist and we are already global."
Labour's Brexit spokesman Sir Keir Starmer said the speech revealed the government's intention to "casually cast aside" rights and protections and ignore the benefits of the EU single market.
"Nobody will be fooled or reassured by the foreign secretary's empty rhetoric," he said.
Conservative MP Sarah Wollaston accused Mr Johnson of an "optimism bias" about the benefits of Brexit.
The SNP's Brexit spokesman Stephen Gethins said he was not reassured by Mr Johnson's speech and said the government "still can't really tell us what leaving the EU will mean".
And Liberal Democrat Tom Brake said the speech was mainly about "Boris' ambitions to become the next prime minister".
How did fellow Brexiteers respond?
Conservative MP Nigel Evans said the foreign secretary had reflected the views of many people, irrespective of how they voted in 2016, that the UK now needed to get on with the process of leaving.
"It's also about the opportunities that are now going to exist for the United Kingdom," he told the BBC.
Former Cabinet minister John Redwood said the speech may have come as a surprise to those who, in his opinion, mistakenly believed the government was set "to reinvent a customs union, forego an independent trade policy and accept the need to follow all new EU laws".
That Boris speech - we know that the Foreign Secretary’s speech today was checked and approved by 10 Downing Street, and is a statement of government policy. https://t.co/LR3FK9h3HL— John Redwood (@johnredwood) February 14, 2018
The BBC's Norman Smith on Johnson's challenge
For many Remainers, Boris Johnson is the bogeyman of Brexit, heartily loathed for his approach and some of his claims during the referendum campaign.
He set himself an ambitious aim of trying to reassure Remainers - but at times it sounded as if we were back in the campaign, which served to highlight just how divisive that debate was.
I was left with the thought that perhaps the person most relieved would be Theresa May, as he repeatedly and doggedly stuck to the principles set out in her Lancaster House speech.
Juncker: No superstate ambitions
Asked about Mr Johnson's speech, European Commission president Jean-Claude Juncker hit back at suggestions from some critics he is seeking to create an EU "superstate".
Mr Juncker replied: "Some in the British political society are against the truth, pretending that I am a stupid, stubborn federalist, that I am in favour of a European superstate.
"I am strictly against a European superstate. We are not the United States of America, we are the European Union, which is a rich body because we have these 27, or 28, nations.
"The European Union cannot be built against the European nations, so this is total nonsense."