She has become one of the most senior figures in the Church of England and the first woman to hold the post.
In her sermon, she spoke of the challenges facing London, particularly escalating knife crime, and referenced historical sexual abuse within the Church.
Bishop Sarah called for a culture that "challenges deference and the abuse of power" and for victims of abuse to be listened to.
She added: "We need to speak up for the whole of London, to work to challenge the violence and the crime that have led mothers to clean their own children's blood from our pavements. Could there be a starker image or a more urgent wake-up call for all who love this city, who believe it can have an even better future?"
She also preached on the theme of "being subversive for Christ" - noting that 105 years ago this week, suffragettes placed a bomb underneath the same seat upon which she had been enthroned as Bishop of London.
The former nurse was invested on International Nurses Day - Florence Nightingale's birthday - echoing the new bishop's former career as a nurse.
Bishop Mullally performed the tradition of knocking three times on the Cathedral's Great West door with her pastoral staff, a spokesman for the Diocese of London said.
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