He said the engine would help North Korea achieve world-class satellite launch capability, it added.
The development, not confirmed elsewhere, comes as US Secretary of State Rex Tillerson visits China - Pyongyang's main ally.
After personally overseeing the test, Mr Kim "emphasized that the world will soon witness the great significance of the epoch-making victory we achieved today", KCNA reported, adding that it marked the birth of the country's rocket industry.
Mr Tillerson's East Asian tour has been dominated by anxieties over North Korea's nuclear capabilities.
In South Korea on Friday, he said a US military response would be on the table if North Korea threatened South Korea or US forces.
The US and China pledged to work together to get the North to take "a different course" and move away from its weapons programmes after Mr Tillerson met his Chinese counterpart on Saturday.
North Korea has conducted five nuclear tests and a series of missile launches. Experts and government officials believe it is working to develop nuclear-warhead missiles that can reach the US.
Kim Jong Un has said the country is close to a test-launch of an intercontinental ballistic missile.
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