Some 2,300 people attended the inauguration at a convention center, where Leung chose to speak in mainland China's Mandarin dialect instead of Hong Kong's Cantonese dialect, The New York Times reported.
One protester stood up during the ceremony and began shouting, but was quickly escorted out by security officials, The Times said.
Outside, several thousand protesters marched through the central shopping district carrying banners and chanting anti-Beijing and pro-democracy slogans.
Since Britain returned the colony of Hong Kong to China in 1997, residents don't vote for their executive leader. Instead a 1,200-person election committee votes among itself.
While Leung is openly supportive of Beijing's communist party philosophies, he also lives in a $64 million estate in an exclusive Hong Kong neighborhood, the report said.
During his campaign, he promised to eliminate the rising trend of mainland Chinese women coming to Hong Kong to have their babies.China has a "one child" per family policy, but by giving birth in Hong Kong, mothers are ensuring their children could have Hong Kong passports in the future, the report said.