She will take up the role at the charity on 2 January, replacing David Johnston, who has led the charity for 10 years.
A number of senior figures have left the commission this year. In May, David Holdsworth, the deputy chief executive, departed for a similar role at the Intellectual Property Office, and Michelle Russell announced in September that she would be stepping down as director of investigations, monitoring and enforcement.
Atkinson joined the commission in 2006 as head of corporate affairs before becoming head of information and communications in 2011 and taking up her current role in 2014.
She is also a trustee for Endometriosis UK and previously served on the boards of the Professional Association for Childcare and Early Years and Womankind Worldwide.
Before joining the commission, she was public affairs manager at Nationwide Building Society and had previously worked for a public affairs consultancy.
In a tweet announcing her appointment, she said her role at the commission had been "an amazing privilege".
She said: "My colleagues are dedicated, expert public servants & totally committed to the purpose of ensuring charities thrive and inspire trust. My successor will be very lucky.
"I’ve been talking for ages about the importance of charity and civil society, the critical role of leadership, and the business case and moral imperative for diversity and inclusion – time now to commit myself to this entirely."
Alan Milburn, chair of the SMF, said he believed Atkinson would "be an outstanding leader of the organisation".
He said: "Sarah brings with her a wealth of experience in the charity world and a deep passion for social mobility.
"Her skills will be essential in helping SMF grow and develop as we seek to break down the barriers that stand in the way of so many young people in our country realising their aspirations."
Helen Stephenson, chief executive of the Charity Commission, said: "Sarah has played a crucial role in setting the new strategic direction of the commission. I am especially grateful for everything she has done to help us start to deliver our strategy and our important responsibility to the public of ensuring that charity can thrive and inspire trust."