Left-wing incumbent Rafael Correa is standing down after leading the South American nation for a decade.
Mr Moreno has 38% of the vote with conservative Guillermo Lasso on 30%, based on 60% of returns.
To avoid March's run-off, a candidate needs 40% and a 10 percentage point difference with the nearest rival.
Mr Moreno, a key figure in Mr Correa's cabinet between 2007 and 2013, had been predicted to win the vote.
Mr Moreno has declared himself the winner without waiting for the official results.
"This revolution nobody can stop," he was quoted as saying by Associated Press.
Meanwhile, Mr Lasso's supporters gathered outside the National Electoral Council, saying the run-off was unavoidable.
Centre-right Mr Lasso is a former banker who wants to create a million jobs.
Among his pledges is one to withdraw asylum rights from Wikileaks founder Julian Assange, who has been holed up in the Ecuadorean embassy in London since June 2012.
Mr Moreno, who became paraplegic after being shot in the back in 1998 and has most recently served as UN special envoy on disability and accessibility, has tried to distance himself from Mr Correa in recent years.
Mr Correa, who was elected in 2007 and is unable to stand again, oversaw an economic boom during his 10 years in power.
But a slump in the price of oil meant the oil-exporting country was unable to continue supporting its poverty reduction programmes, and was criticised for not putting money aside for hard times.
Recent corruption scandals have also cast the ruling party in a more negative light, and some have speculated that Ecuador will follow Peru and Argentina in electing a conservative government.
Voting is mandatory in Ecuador, and 12 million people are expected to have cast their ballot.
For the first time, transgender people were able to vote according to their chosen sex, rather than that with which they were born. Men and women vote in separate lines in Ecuador and there were reports of harassment in previous elections.
Partial official results in Ecuador's presidential election have put ruling party candidate Lenin Moreno in the lead - but he may face a run-off.