The test was apparently timed to coincide with the visit of US Secretary of State Rex Tillerson to Beijing on Saturday, where he warned that regional tensions had reached a "dangerous level".
State news agency said Kim had overseen the operation, and "emphasised that the whole world will soon witness what eventful significance the great victory won today carries", KCNA reported, hinting that the North could use the new engine to launch a rocket to put a satellite in orbit.
Rocket engines are easily re-purposed for use in missiles.
Outside observers say that the nuclear-armed Pyongyang's space programme is a fig leaf for weapons tests.
"The development and completion of a new-type high-thrust engine would help consolidate the scientific and technological foundation to match the world-level satellite delivery capability in the field of outer space development," KCNA reported.
"The leader (Kim) noted that the success made in the current test marked a great event of historic significance as it declared a new birth of the Juche-based rocket industry."
On Saturday, the US and China pledged to work together to address the threat posed by North Korea's nuclear programme.
Mr Tillerson has also visited US allies Japan and South Korea where he said the US would no longer observe the "failed" approach of patient diplomacy with Pyongyang, warning that American military action against the North was "on the table".
The tougher US talk followed two North Korean nuclear tests last year and recent missile launches that Pyongyang described as practice for an attack on US bases in Japan.
The last ground test of a high-powered rocket engine was in September last year, which was also observed by Kim.
Kim at that time hailed the test and called for more rocket launches to turn the country into a "possessor of geostationary satellites in a couple of years to come".
A geostationary satellite must be propelled to an altitude of 36,000 kilometres (22,500 miles) and North Korea is showing off its progress in developing a long-range inter-continental ballistic missile that can reach the US east coast, Professor Yang Moo-Jin of the University of North Korean Studies said.
"The North is hinting strongly that it will soon launch a new satellite rocket" from its Sohae satellite launch site, Yang told AFP.
It may also carry out a secretive intercontinental ballistic missile (ICBM) test using a mobile launcher, he added.
"The test is timed with Chinese President Xi Jinping's visit to the United States," and the anniversary of the founding of North's army, he noted.
Negotiations are under way for Xi's first summit with US President Donald Trump next month in the United States.