Talking to the media after the meeting, Sharif said he held no grudge against Zardari. He said he had made no demand for the President's resignation.
"We have Pakistan's interest in mind, not our own," Sharif said.
Everyone has to work together to resolve the problems faced by Pakistan, he said.
The Charter of Democracy signed by the PPP and PML-N in 2006 will be fully implemented, Sharif said.
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Zardari and Sharif met at the presidency after a luncheon hosted by the President for visiting Chinese Premier Li Keqiang.
The overall situation in the country and issues related to the transition of power figured in the discussions, presidential spokesman Farhatullah Babar said.
The President congratulated Sharif on his victory and said he welcomed the new government.
He also wished good luck to the new government.
Zardari said all political parties and democratic forces should be complimented for the peaceful democratic transition which was taking place for the first time in Pakistan's history.
Speaking to the media after the meeting, Zardari said the presidency will fully cooperate fully with Sharif and that he already considered the PML-N chief the Prime Minister of Pakistan.
In reply to a question on whether the PPP would support Sharif in the election of the Prime Minister in the National Assembly, Zardari replied this matter would be decided by Amin Fahim, who heads the Pakistan People's Party-Parliamentarians.