The protest, by a group calling itself Stop Fossil Fuel Subsidies Australia, was the latest in a series of climate demonstrations targeting famous artworks around the world.
Protesters also scrawled graffiti across the covers of the prints, which were removed for cleaning, but were not damaged, according to the gallery.
"A protest has taken place at the National Gallery of Australia following similar incidents here and overseas," the organisation said.
"The National Gallery does not wish to promote these actions and has no further comment."
Warhol's Campbell's Soup screen prints, produced in the 1960s, remain one of the most recognisable symbols of the American pop art movement.
"Warhol took as themes everyday subject matter that resonated because of its familiar origins ... such as the humble can of Campbell's Soup," the National Gallery of Australia wrote alongside the exhibition.
The protesters said they were targeting the Warhol prints to highlight the "danger of capitalism".
"Andy Warhol depicted consumerism gone mad in this iconic series. And now we have capitalism gone mad," climate activist Bonnie Cassen said.
"Families are having to choose between medicine and food for their children while fossil fuel companies return record profits."
Other climate protesters have recently glued their hands to Goya paintings in Madrid, thrown soup on Vincent van Gogh works in London, and smeared mashed potatoes on a Claude Monet masterpiece in Potsdam, near Berlin.
In October, two climate activists glued their hands to the perspex cover of a Picasso painting in a Melbourne art gallery.
The painting, which was not damaged during the demonstration, was being shown on the final day of the gallery's The Picasso Century exhibition.