In reply, Modi said he has "a wonderful experience of working with you, and I would like to cooperate with you to take India-Japan ties to newer heights from now on".
"I appreciate your inviting me to visit Japan. I look forward to meeting you," Modi said.
In their telephone talk Monday, Abe congratulated Modi on the "historic victory" of the Bharatiya Janata Party (BJP) in the general elections.
He said Japan and India share fundamental values and strategic interests as the two biggest democratic countries in Asia "and I believe Japan-India relationship is blessed with the largest potential for development of any bilateral relationship anywhere in the world", according to a press statement from the Japanese embassy.
He expressed his wish to work closely with Modi "towards further development of the Japan-India Strategic and Global Partnership" and invited him to Japan to continue the annual summit meetings.
Modi thanked him for the congratulatory statement, and remarked that "Japan has made a significant contribution to the development of India".
"I would like to express my deep gratitude to the Japanese people and the Japanese government," the prime minister-designate said.
"Japan and India are traditionally tied with common cultural values, and we are recently promoting our mutual strategic partnership. India is facing various problems domestically and internationally, but I will work to squarely address these problems since we are given a clear mandate by the people of India," the statement quoted him as saying.