The British charity is accused of concealing the findings of an inquiry into claims staff used prostitutes while delivering aid in Haiti in 2011.
Penny Lawrence said she was “ashamed” and takes full responsibility.
The Charity Commission has launched a statutory inquiry into Oxfam – which denies a cover-up – but details of its scope have not yet been released.
The watchdog says it has concerns the charity may not have “fully and frankly” disclosed everything it knew about the claims.
It comes after an earlier meeting with the International Development Secretary in a bid to protect Oxfam’s funding from being cut.
Ms Lawrence joined Oxfam GB in 2006 as international programmes director, leading teams across 60 countries, according to the charity’s website.
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“Concerns were raised about the behaviour of staff in Chad as well as Haiti that we failed to adequately act upon,” she said in a statement.
“It is now clear that these allegations – involving the use of prostitutes and which related to behaviour of both the country director and members of his team in Chad – were raised before he moved to Haiti.”
The allegations emerged in The Times last Friday, which said Oxfam’s country director for Haiti, Roland van Hauwermeiren, used prostitutes at a villa rented for him by Oxfam in the aftermath of the 2010 earthquake.
According to the paper, Oxfam knew about concerns over the conduct of Mr van Hauwermeiren and another man when they worked in Chad before they were given senior roles in Haiti.
Mark Goldring, chief executive of Oxfam GB, admitted failings to act on complaints and when it allowed Mr van Hauwermeiren to move onto another post after allegations were revealed.
The European Commission has said it expects full clarity and maximum transparency from Oxfam, adding that it is ready to “cease funding any partner not living up to high ethical standards”.
The charity’s programme in Haiti received €1.7m in EU funds in 2011.
International Development Secretary Penny Mordaunt said Oxfam had apologised for its “appalling” behaviour and that the government had not been told at the time the allegations involved sexual misconduct or beneficiaries.