Mr Kim took centre stage at a memorial service in Pyongyang's main square a day after his father's funeral.
Kim Yong-nam, formally the number two leader, told a million-strong crowd their sorrow would be turned into strength "1,000 times greater under the leadership of comrade Kim Jong-un".
State TV showed Kim Jong-un surrounded by top government and army officials.
The memorial event appeared to be the Kim dynasty's unofficial handover of power, says the BBC's Lucy Williamson in neighbouring South Korea.
A three-minute silence was also held, after which trains and ships throughout the country sounded their horns.
Kim Jong-il died of a heart attack on 17 December, aged 69, state media said. He had ruled North Korea since the death of his father Kim Il-sung in 1994.
"Respected Comrade Kim Jong-un is our party, military and country's supreme leader who inherits great comrade Kim Jong-il's ideology, leadership, character, virtues, grit and courage," Kim Yong-nam told the massive crowd gathered in Kim Il-sung square.
"The fact that he completely resolved the succession matter is Great Comrade Kim Jong-il's most noble achievement."
Screen grab from North Korea state television shows Kim Jong-un (in black overcoat) bowing at the memorial service in Pyongyang on 29 December 2011 Kim Jong-un (in buttoned black overcoat) bows at the memorial in Pyongyang
A top military official, Kim Jong-gak, also addressed the crowd.