Baghdad, Iraq (CNN) -- Iraq's elections "really went very, very well" and the "Iraqi people deserve a lot of congratulations from us," the United States' ambassador to Baghdad said Monday.
"The election was supported by the overwhelming majority of Iraqi people," Christopher Hill told CNN's "American Morning" program, saying there was "a great deal of support for this political process."
Hill was speaking a day after millions of Iraqis turned out to cast ballots for the country's parliament, despite the ongoing threat of violence.
Militants intent on disrupting the vote carried out dozens of attacks, leaving 38 people dead.
Gen. Ray Odierno, the top U.S. military commander in the country, said most of the casualties came from a single incident when bombers collapsed an apartment building in Baghdad, portraying the overall level of security nationwide as good.
"In the rest of Iraq, it was extremely peaceful," he said. "I was very impressed with the coordination and work done by the Iraqi security forces."
Hamdiya al-Husseini, a commissioner of the Independent High Electoral Commission said Monday that the election turnout was 62 percent nationwide and 53 percent in Baghdad.
The commission plans to announce some initial results Tuesday, he said.
Hill said Washington would work with any democratically elected Iraqi government. "We would like to develop this broad relationship with a key country in the Middle East," he said.
Odierno stood by U.S. plans to reduce the number of American troops in Iraq to 50,000 by September 1.
Despite the risks, voter turnout could reach 55 percent, a senior U.S. official told CNN.
The general feeling on election day was the longing for change, of electing a government that will be able to provide basic services like water and electricity, jobs, and security, a CNN crew posted at a Baghdad polling station reported.
A woman in line to vote there said that her vote was a way of fighting back against acts of terrorism.
Another voter, Ali Abdul Hassan, risked the uncertainty of voting Sunday with his 2-month-old infant. "I want my baby to start voting early," he said.
While there were about 60 security incidents reported throughout Iraq, speaking at the White House Sunday, President Barack Obama downplayed the attacks.
Some violence was expected, Obama said, "But overall, the level of security, and the prevention of destabilizing attacks, speaks to the growing capability and professionalism of Iraqi security forces, which took the lead in providing protection at the polls."
The success of the vote showed that "the future of Iraq belongs to the people of Iraq," he added.
Counting began at polling stations at the end of voting, under the gaze of observers. Ballots will be counted twice to ensure accuracy, according to the United Nations Assistance Mission for Iraq.
The number of ballots cast will be reconciled with the number of ballots issued to each polling station, and in case of a significant discrepancy, the station will be audited, the U.N. said.
It was Iraq's fifth nationwide vote since 2003, but only the second for a full four-year-term parliament.
The last time the country had a national vote was in 2005, when the Sunni Arab population boycotted the elections and the political process. A Shiite-led government emerged and the Sunnis, feeling disenfranchised, went on to form the main part of the insurgency.
The U.N. mission in Iraq called the elections an "important milestone in Iraq's democratic progress," serving to strengthen the country's sovereignty and independence as the United States draws down its military presence there.
Sunday's elections were supposed to happen in January but were delayed because of political disagreements and a delay in passing the election law that paved the way for this vote.
There were 18.9 million eligible voters, casting ballots for 325 seats in the Council of Representatives, as Iraq's parliament is called. The seats represent Iraq's 18 provinces.
Around 6,200 candidates from more than 80 political entities were vying for seats. At least a quarter of the positions -- 82 -- are guaranteed to go to women, and eight more have been allocated for minorities. They include five set aside for Christians and one each for the Shabak, Sabaeans (Mandaeans), and Yazidis.
It was the first parliamentary vote to use an open list, in which the citizens vote for political entities and, if they want, also can vote for candidates within those entities, according to the U.N. assistance mission.
Although open lists complicated the training of election staff and the counting of votes, the system enhances the role of the voter in the election beyond casting a vote simply for a political party, the assistance mission said.
At least 25 percent of the candidates on the ballot list of each political entity were women.
Some 300,000 trained election staff -- mostly teachers, principals, and lawyers -- were on hand at polling sites.
CNN's Arwa Damon, Jomana Karadsheh, Mohammed Tawfeeq and Zain Verjee contributed to this report.