Moreno's election victory Sunday was a relief for Assange, who has been holed up in Ecuador's London embassy since 2012 to avoid arrest.
The socialist president-elect's conservative rival, Guillermo Lasso, had vowed to kick Assange out of the embassy.
But Moreno had some stern words after Assange took to Twitter to celebrate Lasso's loss.
"Mr Julian Assange must respect the condition (of asylum) he is in and not meddle in Ecuadoran politics," he said at a news conference.
As results showed Lasso losing on election night, Assange had exuberantly turned around the right-wing candidate's threat to expel him within 30 days.
"I cordially invite Lasso to leave Ecuador within 30 days (with or without his tax haven millions)," he tweeted -- a reference to allegations the ex-banker has money stashed in offshore accounts.
Assange fled to the embassy to avoid arrest and extradition to Sweden, where he faces a rape allegation.
The 45-year-old Australian, who denies the allegation, says he fears Sweden would send him to the United States to face trial for leaking hundreds of thousands of secret US military and diplomatic documents in 2010.
Outgoing President Rafael Correa, a fiery critic of the US, granted Assange asylum, and Moreno has vowed to uphold it.
Assange's case has returned to the spotlight since WikiLeaks was accused of meddling in the US election last year by releasing a damaging trove of hacked emails from presidential candidate Hillary Clinton's campaign and her Democratic party.
That created an awkward situation for the Ecuadoran government, which responded by temporarily restricting his internet access.
