counting of votes was underway in Pakistan, Singh conveyed India's desire to work with him to chart a new course in Indo-Pakistan relations.
In a letter to Sharif, Prime Minister Singh said, "I look forward to working with you and your government to chart a new course and pursue a new destiny in the relations between our countries. I would also like to extend an invitation to you to visit India at a mutually convenient time."
He said the "people of India also welcome your publicly articulated commitment to a relationship between India and Pakistan that is defined by peace, friendship and cooperation".
Singh, who addressed Sharif as "Dear Mian Saheb" in the letter, extended his "heartiest congratulations" on the PML-N's "emphatic victory".
Talking to the media on Saturday night in Lahore, Sharif said he worked hard for the peace process with India before former military ruler Pervez Musharraf deposed him in 1999.
"We'll pick the threads where we left. We want to move toward better relations with India, to resolve the remaining issues through peaceful means, including that of Kashmir," he said.
During the course of his campaign, Sharif had expressed his keenness to visit India at the earliest.
The PML-N returned to power after millions of Pakistanis braved Taliban threats and violence that claimed some 50 lives to vote in the landmark general election that marked the first transition from one civilian government to another in the country's 66-year history.
The "historic election" was a "significant victory for democracy in Pakistan. The people of India have watched with admiration the people of Pakistan braving violence and strife and turning out in large numbers to affirm their democratic rights," Singh said.
He wrote: "You have received a strong mandate to lead Pakistan towards a stable, peaceful and prosperous future. I wish you all success in meeting the hopes and expectations of your people as you prepare to assume your new responsibilities."
The people and political parties of Pakistan "deserve all credit for strengthening the framework of democracy" by participating in the election despite "enormous challenges", the prime minister said.
"My wife joins me in extending to you and Begum Saheba our best wishes," he said.
External affairs minister Salman Khurshid hoped that India will continue to have good relations with Pakistan under Sharif's leadership.
Sharif served as premier during 1990-1993 and 1997-1999 but was ousted from office before he could complete his term ? once on corruption charges and later because of the coup led by Musharraf.