The Change Britain campaign is being led by Mr Johnson's former Vote Leave colleague, Labour's Gisela Stuart.
The group's main aim is to ensure that PM Theresa May's delivers on her promise that "Brexit means Brexit".
Mr Johnson said the group would make sure that the UK gained control over "laws, borders, money and trade".
His comments came as former cabinet minister John Whittingdale urged Mrs May to speed up the formal process of breaking away from Brussels, rather than waiting until next year to trigger Article 50 - after which both the UK and EU will have two years to negotiate the terms of the split, with any extension having to be agreed by all 28 EU members.
In a recorded a video message, in which he declared his support for the campaign, Mr Johnson said: "On June 23 the people of this country voted to leave the European Union and they voted for change.
"They did so by a clear majority. But there were many people who also voted for Remain.
"So it's absolutely vital that we work together, Leavers and Remainers, as we seize the opportunities that this country now has to forge a positive and exciting new relationship, not just with the European Union, but also with the rest of the world; changing Britain and making it global again.
"Now more than ever we need to show the British people that, as politicians, we are listening to what they have to say.
"Brexit means Brexit and that means delivering on their instructions and restoring UK control over our laws, borders, money and trade."
Ms Stuart said the EU referendum meant the country would undergo "the biggest change in 40 years in our diplomatic, trade and economic relationships".
"It also creates a unique opportunity to review our democratic and constitutional arrangements," she said.
"The referendum also marked a more profound political change than a change of occupancy in Downing Street. It has forced us to acknowledge that people in large sections of the UK have lost faith in political parties and the Westminster elite.
"While millions in this country enjoy unprecedented prosperity and freedoms, many millions more feel despair at their sense of exclusion and diminishing prospects."
Meanwhile, Mr Whittingdale, the former culture secretary, urged Mrs May to push ahead with triggering Article 50, which begins the formal two-year process of leaving the EU.
"Article 50 is the beginning of the process rather than the end," he told the Daily Telegraph. "We do need to get the formal process under way. I don't say that it has to happen tomorrow but I would like it to happen pretty soon, and by that I mean weeks, not months."
And Labour leadership hopeful Owen Smith said he would consider applying to rejoin the EU if he became prime minister after UK had already left.
Foreign Secretary Boris Johnson has thrown his support behind a new cross-party campaign aimed at ensuring the UK leaves the European Union.