Louise Richardson was formally installed at a meeting of the university's ruling body at the Sheldonian Theatre.
"In an increasingly complex world the best may not be those who look and sound like ourselves," she said.
Professor Richardson was previously in charge at St Andrews.
She gave a speech to university dons at the congregation, before a procession to the Clarendon Building.
In the speech Mrs Richardson said: "'How do we ensure that they appreciate the value of engaging with ideas they find objectionable, trying through reason to change another's mind, while always being open to changing their own?
"How do we ensure that our students understand the true nature of freedom of inquiry and expression?"
She also said: "In an increasingly complex world the best may not be those who look and sound like ourselves.
"They may not be those who naturally think of coming to Oxford.
"Those with the greatest potential may not be those who have already attained the most.
"We need to seek them out."
Mrs Richardson was born in Ireland and studied at Trinity College, Dublin, and has degrees from the University of California, Los Angeles (UCLA) and Harvard. She became vice chancellor of St Andrews in 2009. She has also held a senior role at Harvard University.
Oxford University appointed its first head in 1230, and all of its vice chancellors have been men.
Prof Richardson, an expert in security and terrorism, was selected in May by a nominating committee headed by the university's chancellor, Lord Patten as the 272nd vice chancellor in Oxford's history.
Oxford is the latest high-profile university to appoint a female leader. Harvard University and Imperial College London are both headed by women.
The first female vice chancellor of Oxford University has called on students to be open minded and engage with "objectionable" ideas.