It would boost the struggling leftist movement in South America after right-leaning governments recently came to power in Argentina, Brazil, and Peru as a commodities boom ended, economies flagged and corruption scandals grew.
While Lasso has said he would evict Assange from the embassy where Ecuador granted him asylum in 2012 to prevent his extradition to Sweden, Moreno has said he could stay, increasing global interest in Sunday's vote.
No official victor has yet been declared.
The opposition candidate in Ecuador's presidential election, Guillermo Lasso, filed on Monday a case alleging electoral fraud with the Organization of American States (OAS).
Lasso won 28.9 percent of the first round's votes, but was expected to pick up more votes after conservative Congresswoman Cynthia Viteri, who finished third in the first round, threw her support behind him. Correa has said he will move to Belgium, where his wife is from, when he leaves office.
The embassy is being constantly monitored by United Kingdom police ever since Assange slipped into it in 2012 and was granted asylum by the Ecuador government.
Former Vice President Lenin Moreno has claimed a win in the presidential runoff ballot amid calls for a recount from his conservative challenger.
"The revolution has triumphed again in Ecuador", tweeted Correa, who has said he will move to Belgium, where his wife is from, when he leaves office.
Guillermo Lasso, presidential candidate for the CREO political party, greets the crowd during his closing campaign event ahead of Sunday's presidential runoff election in Guayaquil, Colombia, Thursday, March 30, 2017.
But there is one person, holed up in Ecuador's embassy in London for the past five years, who must be watching the elections very closely and very nervously, for the outcome may decide whether he finally gets to go "home" (wherever that turns out to be) to face justice and possible imprisonment.
Lasso demanded a recount after three exit polls showed him winning. But they have until 2020 to bring criminal charges on the more serious allegation of rape.
Earlier, a jubilant Lasso told supporters in Guayaquil that he would free political prisoners and heal divisions created by 10 years of iron-fisted rule by Correa.
Government candidate Lenin Moreno, meanwhile, has promised to keep hosting Assange in the Andean country's embassy - although he warned he may take a tougher stance.
"We're not going to allow it", said Lasso, who called on supporters to protest the results peacefully but firmly.
The vote on April 2 was the second round of voting after Moreno won nearly 40% - 10 points higher than his nerest opponent in Lasso - falling just 0.7% short of winning outright in the first round on Feburary 19.
On his Twitter account shortly after the results became known, Assange took a jab at Lasso's pledge.