Kim, who personally supervised the test, said that "the whole world will soon witness what eventful significance the great victory won today carries", according to KCNA.
The Kim regime has a history of making exaggerated claims and belligerent threats that it can't back up, but the latest boast comes amid heightened tensions in the region.
It said the test was meant to confirm the engine's thrust power and gauge the reliability of its control system and structural safety.
In February of 2016 it said it launched a long-range rocket in defiance of a United Nations ban on tests.
North Korea is banned by the global community from pursuing nuclear and missile programmes but has defiantly ploughed ahead. The Japanese government said three out of four missiles launched from North Korea on March 6 landed in the sea less than 200 nautical miles (around 370 kilometers) from Oga. A New York Times story on Saturday said that Kim remains unmoved by Beijing's economic pressure because he clings to nuclear weapons as the only guarantee of safeguarding his power and family dynasty. The simultaneous firings appeared created to outsmart the Terminal High Altitude Area Defense antimissile battery that the United States is deploying to South Korea, which would have difficulty shooting down four targets at once.
"I do not deny there are several individuals, including North Koreans, involved in the assassination and we will use the legal channels to apprehend them", Khalid Abu Bakar told reporters as quoted by the Bernama news agency.
The KCNA said he checked the technical specifications of the engine and preparations for the test, and gave appropriate instructions before going up to the observation post to give the order for the test to begin.
In contrast to the Sunday reports, which did not go into detail about the engine performance, the North noted then that the September test lasted 200 seconds and that a single engine's thrust reached a force of 80 tons, which means it could lift an object weighing 80 tons for 200 seconds.
The United States and China pledged to work together to get the North to take "a different course" and move away from its weapons programmes after U.S. Secretary of State Rex Tillerson met his Chinese counterpart on Saturday. "China has done little to help!"
Zahid also said the body could not be returned to North Korea without consent from Kim's next-of-kin. We are exploring a new range of security and diplomatic measures.
In a total switch from previous policy, Mr Tillerson said: "Let me be very clear: the policy of strategic patience has ended".