Mr Fox was speaking after talks with the outgoing president, Ernesto Zedillo, during which they discussed potential nominees for senior government positions after power is transferred in December.
He urged his supporters not to seek retribution against the Institutional Revolutionary Party (Pri), which has lost power after 71 years of government, and is blamed by many Mexicans for widespread corruption.
He invited members of the Pri and the center-left Party of the Democratic Revolution (PRD) to join what he said would be "a government of concord".
"The reconciliation is under way," said Mr Fox.
He announced the creation of a National Transparency Commission to probe "past events" such as alleged political assassinations, but said he would "not launch a witch-hunt".
'Moral authority'
Speaking after his meeting with President Zedillo, Mr Fox said that the time had come for calm and for Mexicans to work toward national development.
The new president-elect has promised economic growth and educational improvements, with the creation of millions of new jobs.
Large crowds of people poured on to the country's streets to celebrate his win, while financial markets reacted positively to the result, with shares jumping by over 4% and the peso surging in value.
In an interview with the Associated Press news agency, Mr Fox said: "We will have moral authority and democratic legitimacy."
With most of the votes counted, he had won around 43% of the ballot, compared to 36% for the Pri's Francisco Labastida.
The United States and the European Union have both welcomed the result as evidence of Mexico's commitment to democracy.
A US State Department official said Washington looked forward to working with Mr Fox on key issues such as drugs, immigration and trade.
Free and fair
For the first time, the Pri will not be the single largest party in the Congress.
The BBC correspondent in Mexico City says these have probably been the first really clean elections in Mexico's history, thanks to reforms carried out under President Zedillo.
He says that over the past decade, the government has been forced to introduce democratic reforms in order to be accepted as a member of the international community.
This, he says, opened the way for Sunday's historic opposition victory, which has closed a chapter on the single party system.
Split Congress
Voters have also been electing 128 senators, 500 federal deputies and the head of Mexico City's administration.
Correspondents say that, although it will take some time for the final figures to emerge, it seems the Pan leader is likely to face a divided Congress and Senate.
They say Mr Fox's party will dominate, but it will not have an absolute majority and Mr Fox will have to work with the new opposition if he is to succeed.
The new Congress is scheduled to be sworn in on 1 September.
Leftist presidential candidate Cuauhtemoc Cardenas of the Party of the Democratic Revolution (PRD) appeared to be in third place with around 16%.