She replaced Ms Morgan, who made clear her departure was unwilling by saying she was "disappointed" to lose the job.
The new PM wants more women in the Cabinet and a string of ministers will be hoping for promotion, including Andrea Leadsom, who dropped out of the Tory leadership race on Monday, handing Mrs May the keys to No 10.
Mrs May's move came as sacked Tory colleagues were still reeling from the way she swept key David Cameron supporters from the Cabinet in a far more dramatic reshaping of the top team than expected.
She said: "The new Prime Minister was kind enough to offer me a role but it was not one which I felt I could take on".
Green MP Caroline Lucas denounced the decision to shut down DECC as a "serious backwards step", as it would mean no dedicated minister for climate change at the Cabinet table.
Mrs May's leadership rival, Work and Pensions Secretary Stephen Crabb, also quit "in the best interests of my family".
And Mrs May's former number two at the Home Office, James Brokenshire, entered the Cabinet for the first time as Northern Ireland Secretary, replacing Theresa Villiers.
New British Prime Minister Theresa May on Thursday sacked Michael Gove as Justice Secretary and replaced him with Liz Truss, a day after her surprise appointment of lead "Leave" campaigner and former London Mayor Boris Johnson as the new Foreign Secretary.
Amber Rudd took over Mrs May's former role as home secretary, Eurosceptic David Davis was made Brexit secretary and Liam Fox took up a new post of secretary of state for global trade.
Banbury MP Victoria Prentis also welcomed Mrs May as Prime Minister yesterday, adding: "The coming years are going to be very important for the United Kingdom, and I am glad that we will have a steady hand on the tiller of government".
Other eye-catching appointments on the second day of the formation of Mrs May's Government included former transport secretary Patrick McLoughlin as Conservative Party chairman and Chancellor of the Duchy of Lancaster.
Mr Cameron's former parliamentary private secretary Gavin Williamson was appointed Government chief whip, who described the role as "not bad" as he left Downing Street.
In a message on Twitter, Mr Gove - who launched his own bid for the leadership but was eliminated after finishing third in last week's poll of Conservative MPs - said: "It's been an enormous privilege to serve for the last six years".
May continued to wield the political axe, giving a number of Cameron's ministers and secretaries their marching orders, in some cases to make space in the cabinet for more female ministers.
Mrs May has completed appointing her full Cabinet with the final role to be confirmed being Attorney General Jeremy Wright. Britain's new Prime Minister Theresa May arrived at Downing Street on Wednesday after gaining consent from Queen Elizabeth II. Other ministers made a decision to leave the government.