Low will take up the £150,000-a-year position at the beginning of August. The RNID board will now begin a search for his successor.
Low, who has been in his current post for five years, said he was not intimidated by CAF's recent troubles.
"I'm after a challenge," he said, in his first interview following his appointment. "I've had a lot of success with challenges at RNID. The situation at CAF would have put a lot of people off but it attracted me."
Low said he was attracted by the chance to help the whole of the charity sector after developing a "passion for voluntary endeavour" during his time at RNID and as chair of Acevo - a position he will keep if time permits.
He denied that what he described as CAF's "difficult time with staff turnover" amounted to a crisis. "It is not losing money," he said. "All organisations go through cycles, but CAF is getting itself sorted out and it is a great time to join them and move things forward. It is an organisation with a huge potential."
However, he admitted CAF could be doing "so much more" in terms of working closely with charities and responding to their needs. Of the charity's payroll giving service, he said: "It needs to be enabled to grow and find its place."
Low praised the work of former finance director Tony Rogers, who took over on a temporary basis after the resignation of Stephen Ainger. Rogers will now take up the role of chief operating officer.
"He has done a fabulous job both on the finance and people sides," Low said. "He has been a great, cathartic leader."
A strategic review of the organisation started by Ainger would continue to press ahead, Low said. "The organisation is going through a period of change and it will continue," he said. "The results of the strategic review will have to be implemented. But I am pleased to have the opportunity to work in an organisation with such good people in it."
He added that he would not be "going in with a meat cleaver".
A CAF spokesman said there had been a "strong field of candidates" for the job. Chair Simon Cairns said: "John comes to CAF with a strong background and track record of delivery. My colleagues and I have every confidence he will lead CAF in achieving further milestones in the contribution that the foundation has made and continues to make to donors and charities across the UK and internationally."
The search for a new chief executive at RNID will begin immediately. Low stressed that despite recent achievements such as the RNID's success in improving the quality of NHS hearing aids, there was still a lot for his successor to do. "Nine million people have hearing difficulties in the UK, yet there is still a stigma about wearing a hearing aid," he said.
"I'm leaving RNID because it's a good opportunity for me. I feel RNID is robust and will prosper under fresh leadership."
He said there were several "credible" internal candidates for the RNID job, but that the vacancy would be open for external applications.