State media reported on March 2 that the process of counting the votes had begun. Election officials have said they expect a record turnout.
Early returns show supporters of Supreme Leader Ayatollah Ali Khamenei out in front in many constituencies with the younger sister of President Mahmud Ahmadinejad, Parvin Ahmadinejad, reported to be heading for a loss in their hometown of Garmsar.
Khamenei had called for a high turnout, saying it was a "duty and a right" for every eligible Iranian to vote.
He said a large turnout in the elections on March 2 would send a strong message to Iran's enemies amid the nuclear standoff with the West.
"My advice is the usual advice," he said. "I believe that this [voting] is a responsibility for us, whilst it is also a right that belongs to us. We should make use of this right and undertake this responsibility."
The election is the first major Iranian vote since the disputed reelection of Ahmadinejad in 2009, which sparked mass protests and a government crackdown on the reformist Green Movement.
Meanwhile, Iranian media say the authorities have detained 10 people described as "saboteurs." The Fars news agency quoted Mohammad Naqi Baqeri, head of the election security department, as saying the alleged saboteurs had "come from abroad to Tehran."
More than 3,400 candidates are competing for the 290 seats in the Majlis.
Some 48 million people are eligible to vote.
With the elections being boycotted by Iran's main opposition and reformist groups, the leaders of which have been under house arrest, the vote is seen as a contest between allies of President Ahmadinejad and loyalists backing Khamenei.
Analysts have predicted that candidates aligned with Khamenei will emerge in the dominant position -- dealing a blow to supporters of Ahmadinejad, whose administration has been criticized for failing to revive Iran's sluggish economy, which is suffering from high inflation and unemployment.
The outcome of the election is not expected to result in any change to Iran's foreign policy, as parliament has traditionally played a bigger role in economic policy.
But the result is expected to help set the political stage for the 2013 presidential election, when Ahmadinejad will step down after reaching the end of his two-term limit.
The U.S.-based rights group Human Rights Watch has called the elections "grossly unfair" because most of the approved candidates for the election were conservative regime supporters.
No independent election monitors have been allowed into the country.
Final official results for the elections, which will be counted manually, are expected to be announced late on March 4.
Authorities have urged voters to go to the polls, with state TV, radio, and other media carrying appeals describing voting as a religious duty and warning that low turnout could embolden Western enemies set on attacking Iran.
The June 2009 presidential election and subsequent massive street protests prompted a brutal response from authorities that has included televised mass trials, roundups of dissenters and critics, and allegations of torture and extrajudicial killings.