And she urged leaders to work together to tackle "the big security and human rights challenges of our time".
Mrs May warned that people felt left behind by the "increasing pace of globalisation".
In her first address to the general assembly, Mrs May said:
"We must never forget that we stand here, at this United Nations, as servants of the men and women that we represent back at home.
"And as we do so we must recognise that for too many of those men and women the increasing pace of globalisation has left them feeling left behind.
"The challenge for those of us in this room is to ensure that our governments and our global institutions, such as this United Nations, remain responsive to the people that we serve. That we are capable of adapting our institutions to the demands of the 21st Century."
Mrs May - who also addressed the UN summit on refugees on Monday - said the organisation was "uniquely placed" to tackle war, terrorism, climate change, human trafficking and mass migration.
"The biggest threats to our prosperity and security do not recognise or respect international borders," she said, adding: "And that if we only focus on what we do at home, the job is barely half done."