Greening, who was previously Secretary of State for International Development, was handed the brief after Morgan was sacked.
Elected as MP for Putney in the 2005 election, Greening also previously held the posts of Economic Secretary to the Treasury between May 2010 and October 2011, before becoming Transport Secretary, a position she held until September 2012.
She is the first openly gay or bisexual female cabinet minister in the UK after coming out last month, stating on Twitter: "I campaigned for Stronger In, but sometimes you're better off out!"
Former Education Secretary Nicky Morgan tweeted her disappointment at being sacked.
Disappointed not to be continuing as Education Secretary & Min for Women & Equalities - two wonderful roles it's been a privilege to hold— Nicky Morgan (@NickyMorgan01) July 14, 2016
Junior minister posts are likely to be confirmed later today or tomorrow. Speaking at a reception at the House of Commons to mark Barnardo's 150th anniversary yesterday evening, children's minister Edward Timpson said: "I'd like to keep the job I already have."
It has also been confirmed that responsibility for further education, skills, and apprenticeships, will be brought back under the remit of the Department for Education from the Department for Business, Innovation and Skills.
Meanwhile, Michael Gove, who ran against Theresa May in the Conservative leadership election, has been sacked as Justice Secretary.
He has been replaced by Elizabeth Truss, who, prior to becoming Environment Secretary in August 2014, spent two years as childcare minister.
She takes over the department as a major review into the future of youth justice is set to be published.
Government adviser Sir Martin Narey, who has conducted reviews on social work education, residential care and the adoption system, said he was "dismayed", by the departure of Morgan.
"I very much hope you'll quickly return to government," he added on Twitter.