During the talks, the two leaders have reportedly discussed several key issues such as the situation in the Korean peninsula, and ways to promote economic cooperation between the two countries. The topics also include the process to boost trade and defence cooperation.
Following his extensive talks with Jae-in, Modi said, India is a stakeholder in the peace process in the Korean peninsula and it will make all efforts to ensure peace in the region.
"Credit for initiating the peace process in Korean peninsula goes to President Moon," the prime minister said.
Post their summit, the two countries signed four pacts which will further strengthen their strategic cooperation.
Addressing a joint media event, Modi also showered praise on the South Korean leader for initiating the peace process in the Korean peninsula.
He said by signing an 'early harvest package' document, the two countries have taken a step forward in upgrading their comprehensive economic partnership agreement.
"Proliferation links between East and South Asia are a matter of concern for India," he added.
Earlier on Monday, External Affairs Minister Sushma Swaraj had called on President Moon on Monday and discussed ways to elevate the Special Strategic Partnership between the two countries.
The bilateral summit came after a series of personal and friendly encounters that reflected and promoted the close relationship between the countries on Monday.
"India and South Korea are the world's seventh and 11th largest economies. However, trade between the two countries came to $20 billion last year, which is not small but far short of our expectations," Moon was quoted in a South Korea-India business forum in New Delhi on Monday.
The July 10 summit is the second between Modi and Moon since the South Korean President took office in May 2017. Moon is also scheduled to hold separate talks with his Indian counterpart, Ram Nath Kovind, and attend a state dinner to be hosted by the Indian president.
On Monday, Moon had jointly inaugurated the world's largest mobile phone factory in Noida with Prime Minister Modi. The two leaders also took a Metro ride to Noida while heading for the inauguration of the Samsung facility.
Moon was also treated to a colourful traditional Indian dance performance last night in the national capital. The programme was organised especially for him by Prime Minister Modi.
President Moon and his wife Kim Jung-sook hosted a dinner last night for the Korean community living in India and greeted a group of Indian veterans who had assisted the South during the Korean War.