State media had been calling Mr Kim “the great successor” after the death of his father Kim Jong-il on Monday.
Mr Kim has little political experience and experts believe senior officials are guiding the transition.
“We declare from our hearts … we will complete the task of songun (military-first) revolution under comrade Kim Jong-un”, the paper said in an editorial.
The “songun” policy prioritises spending on North Korea’s armed forces.
The announcement comes on the 20th anniversary of the declaration of the late Kim Jong-Il as supreme commander in 1991.
Large crowds have continued to mourn Mr Kim’s death in the capital Pyongyang, gathering to bow in front of large portraits of him set up around the city.