It put Mr Moon on 41.4%, with his nearest challenger, conservative Hong Joon-Pyo, on 23.3%.
Mr Moon favours greater dialogue with North Korea, in a change to current South Korean policy.
The early election was called after a corruption scandal led to the impeachment of the former president.
Park Geun-hye is accused of allowing a close friend to extort money from companies. She denies all wrongdoing.
If Mr Moon's victory is confirmed, he is likely to be sworn in on Wednesday.