With about two-thirds of precincts reporting, the former South Bend, Ind., mayor received 26.8 percent of the state's delegates, while Sanders, I-Vt., had 25.2 percent. Sen. Elizabeth Warren, D-Mass., had 18.4 percent, former Vice President Joe Biden had 15.4 percent and Sen. Amy Klobuchar, D-Minn., had 12.6 percent.
At the bottom of the field were Andrew Yang (1 percent) and Tom Steyer (0.3 percent).
Buttigieg, speaking on the campaign trail in Laconia, N.H., announced the early results of the caucuses to his supporters.
"We don't know all of the numbers, but we know this much: A campaign that started a year ago with four staff members ... has taken its place at the front of this race to replace the current president with a better vision for this country," he said.
"That fact represents an astonishing victory for this campaign, this candidacy and this vision that you all have been a part of."
The results of the Iowa caucus were delayed by nearly a full day, giving a black eye to the first state voting in the 2020 primary season.
State Democratic Party Chairman Troy Price told the candidates' campaigns more than half of the results would be available by 5 p.m. EST, though he didn't specify when the full results might be known.
Earlier Tuesday, he told reporters the "underlying data collected" on a smartphone app used by precinct leaders to report vote tallies were "sound," but a coding issue in the system produced only partial results.
"This issue was identified and fixed," he said. "The application's reporting issue did not impact the ability of precinct chairs to report data accurately."
The ultimate goal, he said, is to ensure the "integrity and accuracy of the process." He added the party has "every indication" the app was secure and there was no hacking.
"We have staff working around the clock to assist the Iowa Democratic Party to ensure that all votes are counted," he said. "It is clear that the app in question did not function adequately."
The technology vendor must provide "absolute transparent accounting" of what went wrong with the application, he demanded.
Some Iowa precinct chairs had noticed problems with the app during tests earlier Monday but assumed they were sporadic issues, the Des Moines Register reported. The issue turned into a paralyzing malfunction after the caucuses concluded, and organizers found they were unable to log into the app. Price said the trouble was first spotted when initial results began to arrive and officials vetted them with accuracy checks.
Nevada State Democratic Party Chairman William McCurdy II said on Tuesday that the party "can confidently say what happened in the Iowa Caucus last night will not happen in Nevada on Feb. 22, " denying reports that it was set to use the same app.
"We will not be employing the same app or vendor used in the Iowa Caucus," he said. "We had already developed a series of backups and redundant reporting systems and are currently evaluating the best path forward."
Democratic presidential hopefuls who'd spent a number of days in Iowa left for New Hampshire -- the next stop on the primary calendar -- without knowing how they performed. New Hampshire will stage its primary next Tuesday.
"It's a total meltdown," Anita Dunn, an adviser to former Vice President Joe Biden, said earlier.Biden told supporters he'd anticipated "a long night" due to the delay, but added, "I'm feeling good."
"From all indications, it's going to be close," he said. "We're going to walk out of here with our share of delegates. We don't know what it is yet, but we feel good about how we are."
The campaign of Sen. Bernie Sanders, who has led some of the recent polls, released its own internal polling numbers showing him as the winner with 30 percent of the vote in nearly 40 percent of precincts.
"I have a good feeling when [results are announced], we'll be doing well in Iowa," the Vermont senator said.
The Sanders' camp's figures showed former South Bend., Ind., Mayor Pete Buttigieg second with 24.6 percent, followed by 21.2 percent for Massachusetts Sen. Elizabeth Warren and 12.4 percent for Biden. Not all precincts, however, were included in Sanders' numbers.
"We recognize that this does not replace the full data from the Iowa Democratic Party, but we believe firmly that our supporters worked too hard for too long to have the results of that work delayed," Sanders' adviser Jeff Weaver said.
"The race still appears to be wide open, and that's what we thought would happen after last night," Benjamin Gerdes, spokesman for businessman and philanthropist Tom Steyer, told The Hill.
"We still feel like we have a real shot at this thing."
In a speech to her supporters Monday night, Warren focused on criticism of Trump.
"[Tuesday] Donald Trump will make a speech about the state of the union," she said. "But I have a message for every American: Our union is stronger than Donald Trump. And in less than a year, our union will be stronger than ever when that one man is replaced by one very persistent woman."
Minnesota Sen. Amy Klobuchar promised to press forward amid the Iowa mystery.
"We know there are delays, but we know one thing: We are punching above our weight," she told a crowd Monday night. "Somehow, someway, I'm going to get on a plane tonight to New Hampshire."
Buttigieg said he's confident he's the big winner in the Hawkeye State.
"We don't know all the results, but we know by the time it's all said and done, Iowa, you have shocked the nation," he said. "Because by all indications, we are going on to New Hampshire victorious."
"This race is a muddled mess," added candidate Andrew Yang. "That means the opportunity for us is growing because there isn't a clear front-runner or even field. New Hampshire will be more important than ever."
Author Marianne Williamson, who left the race last month, criticized Iowa election officials.
"Making it all about the app is a red herring," she said. "Something went wrong here tonight, and it wasn't just an app.
"In a healthy democracy, [Democratic Party leader] Tom Perez would resign today. As it is, the results will probably never be fully determined, the DNC attitude will be 'They'll get over it,' and the entire incident will hang like a cloud of uncertainty over the entire campaign."