Mrs Coffey, a close friend and ally of Truss, will also be Deputy Prime Minister.
The former Department for Work and Pensions Secretary is considered to be among Ms Truss’s closest political allies and friends, with the pair entering Parliament at the same time in May 2010.
She told awaiting journalists outside Number 10 after being given the role: “I’m very excited thank you”.
Mr Cleverly, who previously held a junior foreign office post, did not stop to talk to journalists after being appointed as the new Foreign Secretary.
Suella Braverman, a one-time leadership candidate earlier in the summer, has been appointed Home Secretary, taking over from Priti Patel, who quit last night. She spoke only to wish journalists a good evening as she left No 10.
Mr Kwarteng takes over the role of Chancellor from Nadhim Zahawi who served over the summer after Rishi Sunak spectacularly resigned, a move which triggered the downfall of Boris Johnson.
Their appointments had been widely expected.
In addition, Wendy Morton, a former minister in the Transport Department, has also been appointed as the Government’s chief whip and will also attend Cabinet, Downing Street said.
Ms Truss began shaping her new-look Cabinet after giving a speech outside Downing Street in which she vowed to create an “aspiration nation”.
Shortly after giving the speech setting out her priorities for Government, Ms Truss began to quickly remove a succession of Cabinet ministers from their positions, including Dominic Raab, Steve Barclay and Grant Shapps.
All three had been prominent backers of Liz Truss’s rival for the top job, Rishi Sunak.
In her address to the nation, Ms Truss outlined her early priorities as growing the economy, tackling the cost-of-living crisis and dealing with NHS backlogs.
She said: “We will transform Britain into an aspiration nation with high-paying jobs, safe streets and where everyone everywhere has the opportunities they deserve.
“I will take action this day and action every day to make it happen.”