It comes as mankind is releasing more emissions into the atmosphere than at any point in history, triggering global weather hazards from heat waves to intense hurricanes to raging wildfires and rapidly acidifying oceans.
Yet the gap between carbon reduction targets demanded by scientists to avert catastrophe and actions thus far taken is only widening.
"I shouldn't be up here. I should be back at school on the other side of the ocean," said Miss Thunberg, 16, who has become the global face of a growing youth movement against climate inaction that mobilised millions in a worldwide strike on Friday.
"You come to us young people for hope. How dare you?" she thundered, her voice at times breaking with emotion.
"We are in the beginning of a mass extinction, and all you can talk about is the money and fairy tales of eternal economic growth. How dare you!"
In a surprising turn of events, President Donald Trump made a brief unscheduled appearance Monday at the UN climate summit, which he had been expected to skip entirely.
Mr Trump, who has repeatedly expressed doubt about the overwhelming scientific consensus on manmade causes of global warming, spent a few minutes in the hall of the General Assembly where he applauded Indian Prime Minister Narendra Modi's speech then left.
Earlier, opening the summit, Secretary-General Antonio Guterres said: "The climate emergency is a race we are losing, but it is a race we can win."
French President Emmanuel Macron invited his counterparts from Chile, Colombia and Bolivia to a meeting where $500 million in extra funds were pledged by the World Bank, Inter-American Development Bank, and non-profit Conservation International to protect the world's rainforests.
Fewer than half the 136 heads of government or state in New York this week to attend the UN General Assembly will be present Monday.
Brazil, Australia absent
Among those absent are President Jair Bolsonaro of Brazil, under whose leadership the Amazon rainforest is continuing to burn at record rates, and Australian Prime Minister Scott Morrison whose government has pursued an aggressively pro-coal agenda.
China, the world's biggest carbon emitter by far but also a leader in renewables, was represented by foreign minister Wang Yi who called on developed countries to lead by example in reducing emissions - but also said China was respecting its climate change promises.
Laurence Tubiana, CEO of the European Climate Foundation and one of the architects of the Paris agreement, told AFP the summit represented a "moment of truth" but was also taking place at a time of great political headwinds.
"There's a tension between the countries that want to go ahead to translate their goals into real policies" and those that do not, she said.
"We can hope for the best, that this group of progressive countries and actors and local authorities prepare for the second wave, to demonstrate that this is where modernity is, where progress is, and even where economic growth could be."
In his speech, Mr Macron made a clear reference to Thunberg and the other young speakers who preceded him. "No official can remain deaf to this demand for intergenerational justice," he said.
"We need this youth to help us change things... and put more pressure on those who do not want to move."
He also lauded Russia, which ratified the Paris agreement on Monday, and said Europe must do more, repeating a vow to close coal-fired plants by 2022.
German Chancellor Angela Merkel, who has been criticized by activists for not doing enough, said it was her government's responsibility to "take everyone along with them," including those who doubt climate change.
Mr Guterres has asked countries to bring "concrete, realistic plans" to enhance commitments made in 2015 in Paris toward the goal of limiting long-term warming to less than two degrees Celsius - and ideally 1.5 degrees Celsius - over pre-industrial levels.
These are deemed important to avoid hitting a number of so-called "tipping points," like the melting of polar permafrost, that could trigger irreversible warming and fundamentally alter weather events and ecosystems.
But officials have been careful to manage expectations and say Monday's summit is also a run-up event to the 2020 UN climate summit that the UK will host in Glasgow.