During a half-hour telephone conversation, Trump said the two leaders discussed topics including border security, trade and the North American Free Trade Agreement, adding that "I think the relationship will be a very good one".
"Congratulations to Andres Manuel López Obrador, the newly elected president of Mexico". "There is much to be done that will benefit both the United States and Mexico!"
Lopez Obrador said he planned a meeting with worldwide leaders - including Pope Francis, rights activists and the United Nations - for Tuesday night to discuss ideas to solve Mexico's violent crime.
Lopez Obrador spoke to the crowd there around midnight, saying to loud applause: "Today one stage has ended and we are going to begin another one - we are going to transform Mexico!" That is what we always want to maintain with the USA government, that there be mutual respect.
Urzua said the new government was open to proposals that Mexico create a fiscal council to help control deficits and budget spending.
An official quick-count revealed that Lopez Obrador took the lead with 53-53.8% of votes, according to the electoral authority.
Governing party candidate José Antonio Meade, Institutional Revolutionary Party (PRI) 14.8 percent. In his first loss - by a mere 0.56 percent to conservative Felipe Calderon in 2006 - his supporters held months-long protests in Mexico City and he referred to himself as "the legitimate president".
Lopez Obrador has promised a "transformation" of Mexico and pledged to give scholarships or paid apprenticeships to youth, and increase support payments for the elderly.
But the president-elect's senior finance official, Carlos Urzua, tempered fears by clarifying that the Lopez Obrador government was not planning on pursuing major legislative overhauls of the oil sector.
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Then, he cried fraud and declared himself the rightful victor, but alienated many supporters with huge street protests that brought much of the capital to a standstill for weeks. He said he will govern for the poor and fight rampant corruption, and also pulled back from promises to completely scrap energy and education reforms instituted by the Pena Nieto administration.
Lopez Obrador's team will study labour costs as well as clamping down on corruption in Mexico's largest company.
Mexican's President-elect has been a vocal critic of Donald Trump in the past.
"We will search for a relationship with the USA government based on friendship and development, always inculcated with mutual respect and in defense of our compatriots, immigrants who live and work in that country", he said.
Last year, Mexico recorded more homicides than at any point in its modern history, and it is on track to break that record this year.
"The fundamental problem is that this country has a huge demand for drugs", she said. "We are overweight local currency government bonds and the Mexican peso".
Compared with his predecessors, Lopez Obrador is likely to be more focused on domestic economic issues than on settling trade issues with the United States, O'Neil said.
"In the case of Mexico, obviously we share a border with them (and) this president has made very clear about building that wall and having Mexico pay for it", the aide, Kellyanne Conway, said on Fox News.
Carlos Urzua, Lopez Obrador's pick to be his finance minister when he takes office on December 1, told Reuters on Monday that the new government will aim to reform the pension system and to reduce costs at Pemex during its six-year term.