Mandela was a surprise guest at the closing ceremony of the event in Johannesburg, which brought together 1,300 young delegates to debate world issues.
Dublin has been selected as the host venue for next year, with bosses hoping to secure appearances from Kofi Annan, and Bill or Hillary Clinton.
As the 2013 summit drew to a close, Mr Quinn invited delegates to the next meeting.
"Dublin is a cosmopolitan city, blessed by the presence of young people from every corner of the globe, enriched by their energy and their ideas," he said.
"The Irish have always been a global people, rooted in their locality, but reaching out to the world. Dublin's a good place for the summit."
Mr Quinn also urged the attendees to become "disruptive leaders". "It is about young people taking action and making sure they are not sleep-walked into the agenda of others," he said. "It will be about young people acting locally, but thinking globally.
"Together we can shake the world up ane be creative to solve challenges."
In her speech, Winnie Mandela admitted her generation had failed young people.
"We could have done better," she conceded. "Comrade [Nelson] Mandela says there can be no greater reflection of a society than how it treats its children and here you are creating a world that we should have created for you.
"If we are to pay respect to our beloved Madiba [Nelson Mandela], then use us as you champion your way forward. You are the torch bearers of a future economy that empowers the youth."
Dublin hosts the summit on its fifth anniversary after being chosen in a contest between it, Hong Kong and Edinburgh.
Bob Coggins, one of those behind the bid, told the Herald he wanted to replicate the buzz around the Special Olympics and Paralympic Games. "We want to pull together the best elements of Irish life and culture, to create the ultimate Irish welcome and create an inspirational summit," he said.