Writing in The Times, the Rotherham-born former party leader, who served as Foreign Secretary from 2010 to 2014, said "a world without agreed responsibilities on climate change is a world of border carbon taxes on many imports, trade agreements in doubt and intensifying anger as time for vital action begins to run out".
Mr Hague said he was offering new Foreign Secretary Ms Truss, who was raised in Leeds, some advice as "an old-timer who wishes her well".
He said that among the multiple challenges facing her in the new role will be that of dealing with the consequences of climate change.
"The importance of the world reaching agreement at Cop26 is widely understood, but the dire consequences for international relations of failure are not yet appreciated," said the former Richmond MP.
"A world without agreed responsibilities on climate change is a world of border carbon taxes on many imports, trade agreements in doubt, and intensifying anger as time for vital action begins to run out.
"What actions will we take against countries that still burn coal? Will we trade with nations that cut down their forests? The climate is at stake — but so are friendly relations between peoples, continents and states."
It comes after Boris Johnson said this week that the Cop26 climate change summit in Glasgow this year was “a turning point for the world”.
“It’s the moment when we have to grow up and take our responsibilities,” the Prime Minister said.
“We go through a period of glorious indifference about the world, we have been through that, we have been through our childhood, if you like.
“We have now got to realise this is a problem that requires grip and there are changes we are going to have to make.
“But people should be optimistic because we can make these changes whilst encouraging the creation of literally millions of high-wage, high-skilled jobs.”
Mr Hague added in his column that Ms Truss has a "great opportunity" despite the many challenges of her new post given it is being combined with her work as minister for women and equalities.
"Around the globe, the countries that least respect the role and rights of women are those most characterised by poverty, conflict or tyranny. By being an effective leader for girls’ education, preventing sexual violence and advancing the leadership of women, Liz Truss has the chance to advance one of the most vital causes of the 21st century and to show what Britain can do."