During an economic summit in Vladivostok, Russia, Mr Moon urged Mr Putin and Chinese President Xi Jinping to play a role in stopping North Korea's nuclear and missile provocations.
President Moon Jae-in, who is visiting Vladivostok to attend the Eastern Economic Forum, emphasized that South Korea is the best partner for Russia's Far East development.
Abe condemned Pyongyang's nuclear tests.
The North's Sunday test, which was its largest nuclear detonation, was met with worldwide condemnation, and drew a stern rebuke from Defense Secretary James Mattis who warned that any threat to the USA or its allies "will be met with a massive military response - a response both effective and overwhelming".
He told reporters before his departure from Japan that "We must make North Korea understand there is no bright future for the country if it pursues the current path". The full deployment of the THAAD, which consists of at least six launchers, comes 14 months after Washington and Seoul in July 2016 agreed to install it on South Korean territory to counteract the North Korean threat.
North Korea on Sunday detonated a hydrogen bomb that can be carried by an intercontinental ballistic missile (ICBM), North Korea's Central Television announced.
Putin said he and Abe discussed the prospect of a peace treaty ending World War Two hostilities.
"When a rogue regime has a nuclear weapon and an ICBM pointed at you, you do not take steps to lower your guard".
Peter Symonds CorrespondentThe North Korean nuclear test yesterday, its sixth and most powerful, has once again exposed the extremely volatile and precarious state of global geopolitics and the great danger of a descent into a nuclear world war.
The next day, however, Defense Ministry spokesman Moon Sang-gyun said there was no change in the administration's policy on the denuclearization for the Korean Peninsula.
And, at a Conference on Disarmament in Geneva on Tuesday, North Korea's ambassador to the United Nations called Sunday's nuclear test a "gift package" for the U.S. - and warned there would be more.
The concerns about North Korea and its nuclear test on September 3 apparently led the South Korean side to take a more cooperative tone on historical issues that exist between Tokyo and Seoul. American strategists regard domination of the vast Eurasian land mass as the key to USA global hegemony and China as the chief obstacle to that goal.
The U.N. Security Council is now debating imposing tougher sanctions on North Korea, with Japan and the United States pushing for an oil embargo on the reclusive state.
Putin also called for the re-implementation of worldwide law and dialogue among concerned parties over the North Korea issue.
The two leaders also confirmed that every effort must be made to persuade China and Russian Federation to go along with a petroleum trade ban being proposed against North Korea.