Johnson had resigned, saying the Brexit "dream is dying, suffocated by needless self-doubt".
Speaking before Mr Johnson's resignation Mr Dunne said the Prime Minister should be allowed to get on with her work.
British Prime Minister Theresa May has met with her Cabinet as she tries to restore government unity after the resignations of two top ministers over Brexit.
"Top-level cabinet resignations usually spell trouble for a government and Boris Johnson's might well have led to a leadership challenge, but May appears to have emerged unscathed from a meeting of her party, for now", said Al Jazeera's Jonah Hill, reporting from London. He also said he will "step up the preparations for "no deal" in case the spirit of pragmatism and compromise that we have extended is not reciprocated".
"The Government have no doubt that this support will continue", it said.
Jeremy served as attorney general for England and Wales since 2014 but will now turn his attention to media, culture, and all things related to the development of digital radio.
May, the leader of the Conservative party, believed she had secured a hard-won agreement with her deeply divided cabinet of ministers on Friday to keep the closest possible trading ties with the EU.
In his resignation letter to May, Johnson accused her waving "white flags" of surrender in negotiations with European Union leaders.
New Brexit secretary Dominic Raab also vowed to strike the "best deal with our European Union friends and partners" while preparing for an increasingly likely no-deal scenario.
If she chose to fight, she would need the support of more than 50% of Conservative MPs - now 159 - in the confidence vote to stay in office. In her statement to members of Parliament, May acknowledged the splits in her government and the departures of her ministers. He said: "I think anyone who is thinking about a potential leadership challenge should go and buy themselves a coffee and think about new ways of supporting the prime minister".
"This mess is all of the Prime Ministers own making", opposition Labour Party leader Jeremy Corbyn told lawmakers.
"What I want to say to them is Britain is going to be a dependable ally, a country that stands up for the values that matter to the people of this country, and will be a strong confident voice in the world".