Singh wrote a letter to Waheed, who took over after Mohamed Nashed stepped down in the face of widespread protests.
"I take this opportunity to extend my warm felicitations to you on your assumption of office as President of the Maldives," he said in the letter.
The Prime Minister said that as a "close and friendly neighbour" India will, "as always, continue to support the Maldivian people`s efforts to build a stable, peaceful and prosperous country."
Noting that the two countries share a common destiny and have common security interests, he told Waheed that "India is committed to working with you and the Government in Maldives to further enhance our close, bilateral cooperation to mutual benefit and for the continued security, progress and prosperity of our two countries."
Nasheed announced his resignation after weeks of public protests over his order to arrest a senior judge, who had ordered release of a government critic, and handed over the power to his Vice President Mohammed Waheed Hassan.
India had yesterday described the change of regime in Maldives due to protests as an internal matter of the country.