Prime Minister Theresa May announced Hunt's new role in a tweet, adding that the queen had approved the change.
Jeremy Hunt, who has been health secretary for the past six years, was a Remain campaigner in the 2016 European Union referendum.
Mr Davis handed in his resignation late on Sunday night over the Prime Minsiter's plans to maintain a "common rulebook" with the EU on goods after Brexit, effectively keeping one foot in the customs union and remaining tied to Brussels regulations and ruling out free trade deals with third countries.
Dominic Raab has been appointed the UK's new Brexit minister after the resignation of David Davis. "That dream is dying, suffocated by needless self-doubt".
But ministerial aide Chris Green resigned his position as a parliamentary private secretary to the Department for Transport after the meeting, saying in a letter to Mrs May that she had confirmed his fears that "we would not really leave the EU" under her proposals.
However both Johnson and Davis expressed they were not fully on board with the plans.
With Britain due to leave the 28-nation bloc on March 29, 2019, European Union officials have warned Britain repeatedly that time is running out to seal a deal spelling out the terms of the divorce and a post-split relationship.
The two departures seemed to shatter May's own proclamation of cabinet unity last Friday, when she said she believed she had, after two years of wrangling, secured agreement on Britain's biggest foreign and trading policy shift in nearly half a century.
Mrs May told MPs that the plan agreed by the cabinet at Chequers was the basis of a "responsible and credible" offer to restart renegotiations with the EU.
Katie Bylis continued: 'As for the ongoing Brexit negotiations, it's now evens that no deal will be reached by the March 2019 deadline and if any other key players leave the Tory Party then those odds will shorten even further'.
Yet he and Johnson have shared a mutual affinity for one another, so his departure may put the already tenuous U.S. -U.K. relationship on even shakier ground, especially if May enters into conversations with Trump from a weakened position.
"In the two years since the referendum, we have had a spirited national debate, with robust views echoing around the cabinet table as they have on breakfast tables up and down the country", she said.
The likelihood of a Conservative leadership contest now looms.
Earlier in the day the EU Commission had "no specific comment" on the overnight resignation of Davis, which it said was "not a problem for the EU". "I believe that is what people voted for when they voted to leave and we will deliver in faith with the British people", she said.