Canon Penberthy was nominated as Bishop of St Davids at the start of November and is due to take up the role in January.
She will be the first woman to be appointed as a bishop in Wales.
Domestic violence in England and Wales
4.6 million women have experienced domestic abuse since the age of 16
On average, 2 women a week are killed by partners or ex-partners
Domestic violence costs Wales £303.5m annually
Source: Welsh Women's Aid
In an interview with the BBC show The Wales Report, Canon Penberthy said that legislation "won't be enough" to end domestic violence and that everyone must play their part.
She said: "It's hard to believe that in 2016, domestic violence is widespread and women are frightened to speak out.
"Unfortunately it's still the case that a culture that sees abuse which happens behind closed doors is somehow acceptable.
"The phrase 'it's only a domestic' is still too often heard."
Her comments coincide with the UN's 16 Days of Activism Against Domestic Violence Campaign, which began on 25 November.
According to Welsh Women's Aid, 4.6 million women in England and Wales have been victims of domestic violence.
Canon Penberthy was among the first women to be ordained as a priest in Wales in 1997.