"Environmental degradation, climate change and unsustainable development constitute some of the most pressing and serious threats to the ability of present and future generations to enjoy human rights, including the right to life," the resolution said.
David Boyd, the UN special rapporteur on human rights and the environment, called the decision a "historic breakthrough".
"This has life-changing potential in a world where the global environmental crisis causes more than nine million premature deaths every year," he said.
The text - proposed by Costa Rica, the Maldives, Morocco, Slovenia and Switzerland - was passed with 43 votes in favour and four abstentions from Russia, India, China and Japan, prompting a rare burst of applause in the Geneva forum.
The UK, which was among the critics of the proposal in recent intense negotiations, voted in favour in a surprise last-minute move.
Its ambassador to the UN in Geneva, Rita French, said the UK was voting "yes" because it shared supporters' ambition to tackle climate change but added that countries would not be bound to the resolution's terms.
The United States did not vote since it is not currently a member of the 47-member Council.
Costa Rica's ambassador, Catalina Devandas Aguilar, said the decision will "send a powerful message to communities around the world struggling with climate hardship that they are not alone".
Critics had raised various objections, saying the Council was not the appropriate forum and citing legal concerns.
John Knox, a former UN special rapporteur, said ahead of the vote that those who had criticised the resolution were "on the wrong side of history".
The World Health Organisation estimates that 13.7 million deaths a year, or around 24.3 per cent of the global total, are due to environmental risks such as air pollution and chemical exposure.
Another proposal led by the Marshall Islands to create a new special rapporteur on climate change was also approved by the Council on Friday.