"As program director at the time, I am ashamed that this happened on my watch and I take full responsibility," Penny Lawrence said in a statement.
Oxfam has faced severe criticism for a lack of transparency over misconduct allegations against staff members accused of using prostitutes in Haiti following a devastating 2010 earthquake.
The charity has denied covering up the scandal.
Oxfam's chair of trustees Caroline Thomson and chief executive Mark Goldring were called to explain themselves at a meeting on Monday with International Development Secretary Penny Mordaunt.
The minister said in a statement that she had issued a letter to all charities working overseas to demand that they "do more... so that we have absolute assurance" in their moral leadership.
Mordaunt also announced plans to tackle sexual exploitation and abuse "across the UN and other international organizations."
Thomson described the meeting as "very challenging but constructive."
"We recognize that we have some way to go to persuade her that we have the right moral leadership to be fully entrusted with public money."
"But we are committed to working with her, DFID and the Charity Commission to prove we can meet her expectations," she said in a statement, referring to the Department for International Development.
Oxfam received £31.7 million (35.7 million euros, $43.8 million) from state coffers last year, and Mordaunt warned that could be under threat, saying "we will not work with any organization that does not live up to the high standards on safeguarding and protection that we require."
The EU on Monday also said that the bloc would withhold funding if charities breached ethical standards.
"We expect Oxfam to fully clarify the allegations with maximum transparency as a matter of urgency, and we're ready to review and, if needed, cease funding to any partner who is not living up to the required high ethical standards," EU spokeswoman Maja Kocijancic told reporters.