Johnson also urged Brexit backers to "build bridges" with those who voted for Britain to stay in the EU after Thursday's bitterly divisive referendum and played down the prospects of Scottish independence.
"The only change - and it will not come in any great rush - is that the UK will extricate itself from the EU's extraordinary and opaque system of legislation," Johnson wrote in his column for the Daily Telegraph.
"British people will still be able to go and work in the EU; to live; to travel; to study; to buy homes and settle down," he said.
Britons voted by 52 per cent to 48 per cent in favour of leaving the European Union in a vote that sent shockwaves through the financial markets, toppled Cameron and put world leaders in crisis mode.
"There will still be intense and intensifying European cooperation and partnership in a huge number of fields: the arts, the sciences, the universities, and on improving the environment," he said.
Johnson also urged Brexit supporters to "build bridges" with Britons who had supported staying.
"They are our neighbours, brothers and sisters who did what they passionately believe was right.
"We must reach out, we must heal, we must build bridges - because it is clear that some have feelings of dismay, and of loss, and confusion," he wrote.
"At home and abroad, the negative consequences are being wildly overdone, and the upside is being ignored," he said.
He also reacted to Scottish First Minister Nicola Sturgeon's statement that a new independence referendum within two years was now "highly likely" because most Scots had voted to stay in the EU.
"We had one Scotland referendum in 2014, and I do not detect any real appetite to have another one soon.
"It goes without saying that we are much better together in forging a new and better relationship with the EU - based on free trade and partnership, rather than a federal system," he said.
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