each man extended an invitation to the other to visit, a diplomatic nicety in some parts of the world, but a heavily symbolic step for South Asia's arch-enemies.
Asked by an Indian journalist if he would invite Singh for his swearing-in as prime minister, he said: "I will be very happy to extend that invitation."
"There are fears on your side, there are fears on our side," Sharif added during a news conference at his home on the outskirts of Lahore.
"We have to seriously address this."
Sharif's power base is Pakistan's most prosperous province, Punjab, which sits across the border from an Indian state that shares the same name.
A free-marketeer, he wants to see trade between the two countries unshackled, and he has a history of making conciliatory gestures towards New Delhi.
In 1999, when he was last prime minister, Sharif stood at the frontier post waiting to welcome his counterpart - Atal Behari Vajpayee - to arrive on the inaugural run of a bus service between New Delhi and Lahore.
It was a moment of high hope for two countries that were divided amid bloodshed at birth in 1947 and went to war three times in the decades that followed.
But by May of that year, the two sides were sucked into a new conflict as the then-army chief of Pakistan, Pervez Musharraf, sent forces across the line dividing the disputed Himalayan region of Kashmir.
And by October, Sharif had been ousted by Musharraf in a bloodless coup.
Sharif's return to power 14 years later has raised concern that he will again cross swords with the military, which has long controlled the country's foreign and security policies.
But Sharif said he "never had any trouble with the army", just Musharraf, and as prime minister he would ensure that the military and the civilian government work together on the myriad problems facing the country.
Musharraf resigned as president in 2008 and went into self-imposed exile abroad. He returned in March to run in last Saturday's elections.
Instead, he was arrested for his crackdown on the judiciary during his rule and put under house arrest.
Open to like-minded allies
Sharif said his Pakistan Muslim League - Nawaz (PML-N) won enough of the 272 National Assembly seats contested in the election to rule on its own, but suggested he was open to allies joining his government.
"I am not against any coalition. But as far as Islamabad is concerned, we are ourselves in a position to form our own government," Sharif told the news conference.
"All those who share our vision, we will be happy to work with them."
The election was a democratic milestone in a country ruled by the military for more than half its history, marking the first transition from one elected government to another.
However, Sharif inherits a host of challenges from the government led for the past five years by the Pakistan People's Party (PPP), which failed to tackle corruption, poverty and a Taliban insurgency.
Another bailout from the International Monetary Fund to avoid a new balance of payments crisis is seen as inevitable.
Sharif has suggested that he would be willing to implement politically sensitive reforms to secure an IMF lifeline.
He has picked senator Ishaq Dar as his finance minister in the new cabinet, a party spokesman said on Monday.
Dar, who served as finance minister in a previous Sharif cabinet in the 1990s, has said he plans to push provincial governments to collect agricultural taxes, a policy that could set him on a collision course with some of the Pakistan Muslim League's (PML-N) wealthy backers.
Sharif said ahead of the election that Pakistan should reconsider its support for the US war on Islamist militancy and suggested he was in favour of negotiations with the Taliban.
Pakistan backed American efforts to stamp out global militancy after the Sept. 11, 2001, attacks on the United States and was rewarded with billions of dollars in US aid.
But many Pakistanis have grown resentful, saying thousands of Pakistani soldiers have died fighting "America's war".
As prime minister-elect, Sharif chose his words carefully on Monday, saying Islamabad and Washington have "good relations" and "need to listen to each other".
Asked about US drone strikes against militants on Pakistani soil, which many see as a violation of sovereignty, he referred to it as a "challenge" to sovereignty.
"We will sit with our American friends and talk to them about this issue," he said.