The painting, entitled "Flowers" by Italian artist Paolo Porpora, dates back to the 1600s and is part of a collection of 55 artworks on show in the country's capital.
Video footage released by the organizers shows the boy on Sunday trip over a platform in front of the artwork and then brace himself against the painting to break his fall. He then looks around helplessly before walking away.
"The child fell and pressed onto the painting, putting a fist-sized tear in it," an employee at TST Art of Discovery, which organized The Face of Leonardo exhibition in Taipei, told AFP.
The organizers have decided not to seek damages from the boy's family, according to Central News Agency.
The painting is being restored on site, the exhibition's organizers said in a Facebook post Monday.
A self-portrait by Leonardo Da Vinci worth €200 million ($231 million) is also being exhibited at the show, according to the exhibition's website.
Other museum accidents include when a British man in 2006 smashed a set of 300-year-old Chinese vases after tripping over his shoelaces at a museum in Cambridge, and when gaming tycoon Steve Wynn accidentally poked his elbow through the canvas of a $139 million Picasso painting.