In the letter, His Holiness the Dalai Lama said: "Today, the world is passing through difficult and challenging times".
"Peace and security seem under threat in many places. We take courage from the strength of democracy in the Republic of Korea," the Nobel Peace Prize Laureate wrote.
"Now more than ever it is important that all who wish for peace stand together," said His Holiness the Dalai Lama, who is known throughout the world as a leading advocate for world peace and non-violence.
"In seeking to resolve what may seem like intractable problems, the most appropriate way forward is to engage in dialogue in a spirit of understanding, respect and mutual benefit," said the spiritual leader of Tibet who also seeks a "Genuine Autonomy for the Tibetan people within the scope of the Constitution of the PRC through peaceful means.
His Holiness concluded by wishing the President-elect every success in meeting the challenges that lie ahead.
The newly elected president vowed to play a more assertive role in resolving the North's nuclear crisis through dialogue, saying that he was willing to meet with the communist dictator, Kim Jong-un, if the circumstances were right.
Moon also pledged to strengthen the alliance with US, expressing an eagerness for an early summit meeting with President Trump, whose military posturing and diplomatic overtures toward the North Korea in recent weeks have both rattled and reportedly confused many people in South Korea.
But Moon also hinted at balancing diplomacy between the US and China, his country's largest trading partner, over the contentious deployment of an US missile defense system in the South.
Trump called Moon hours after he was formally sworn in on Wednesday. The two leaders agreed to maintain a strong alliance and cooperate in dealing with North Korea's nuclear and missile threats, Mr Moon's office said. They also agreed to hold a summit meeting in US at the earliest opportunity, it said.
Media reports said that Moon is widely expected to introduce a modified version of Mr. Roh's so-called sunshine policy of engaging North Korea with dialogue, humanitarian aid and joint economic projects.
Moon's victory on capped months of political turmoil marked by the impeachment, ouster and arrest on corruption charges of his predecessor, Park Geun-hye. The country has been led by an acting president since Park was removed on March 10, and Moon took office without a customary two-month transition.