"This museum is a testament to resilience of the human spirit," said Bloomberg, the memorial foundation's chairman.
Giuliani, who was mayor when the attack occurred, said what happened remains impossible to comprehend.
Obama also recounted the story of Welles Crowther, a 24-year-old World Trade Centre worker who became known as "the man in the red bandana".
Crowther, a volunteer fire-fighter, led other workers to safety before the south tower collapsed.
One of his red bandanas is now enshrined in the new museum.
