Switzerland, involved in the project from the beginning, said the body would pursue efforts to ensure it is more effective than the highly politicised commission it replaces.
Calmy-Rey described the challenge facing the council as an "immense task," adding that the arrival of the new institution reinforced Geneva’s role as the capital of humanitarian efforts and human rights.
"This event gives cause for much hope, but a lot remains to be done to make this council really efficient and effective," she said. "We have provided a framework [...] I believe that with the council, we have somewhere to talk and not to confront each other."
Calmy-Rey promised that Swiss diplomacy would continue to play a role in the project and saw Switzerland as "giving boosts and providing ideas".
The foreign minister warned that the fight against terrorism must not override fundamental rights. "For Switzerland, any strategy [...] must take into account respect of human rights," she told delegates.
The UN secretary-general, Kofi Annan, also addressed the council on Monday, saying it offered the global body a "great new chance" to renew its struggle for human rights.
Annan said the council should avoid the political point scoring of its discredited predecessor, the Human Rights Commission.
"This council represents a great new chance for the United Nations and for humanity, to renew the struggle for human rights," he added. "I implore you, do not let the opportunity be squandered."
The now-defunct Human Rights Commission was discredited in recent years because some countries with terrible rights records used their membership to protect one another from condemnation.
Stronger body
The Office of the UN High Commissioner for Human Rights said the council should "build on the recognised strengths of its predecessor and flesh out the features that make it a stronger and more effective human rights body".
Fears remain however that the council may prove just as toothless as the commission it replaces as nations accused of rights violations keep their seats.
Cuba, Saudi Arabia, China and Russia all won seats despite their poor human rights records, although others - notably Iran - were defeated.
"There are countries on the council that have pretty bad human rights records," said an Amnesty International spokesman. "But the council has to reflect UN membership. They reflect the reality of the world."
The Swiss are preparing a document on the new institution’s operating procedures, including how it should carry out human rights reviews of all 191 UN member states and how often.
This is to be discussed at a conference in Lausanne on August 28.
Switzerland will also assume responsibility for part of the expansion of the Office of the UN High Commissioner for Human Rights, whose budget and staff are doubling.
Switzerland plans as well to provide offices for the 35 countries that do not have permanent missions in Geneva. To date, 15 countries have taken advantage of this.