Both Mr. Sanders, of Vermont, and Mr. Buttigieg, the former mayor of South Bend, Ind., have tried to claim an overall victory in the state, based on their internal campaign data, and they were too closely matched in the initial returns to declare a single winner. But in New Hampshire on Tuesday, Mr. Buttigieg brandished the first wave of data to once again declare something like victory.
“They’re not complete, but results are in from a majority of precincts and they show our campaign in first place,” he said.
The strength displayed so far by Mr. Sanders and Mr. Buttigieg underscored just how fluid the Democratic race remains, with a 78-year-old democratic socialist who ran for president four years ago deadlocked in Iowa with a 38-year-old former municipal official who was largely unknown a year ago — and both of them substantially ahead of a former two-term vice president.
The result also underscored the divisions on the moderate side of the Democratic Party, with three candidates closer to the political center — Mr. Buttigieg, Mr. Biden and Ms. Klobuchar — collecting a majority of delegate votes so far, but a candidate of the left, Mr. Sanders, still in a position to prevail because of the strength of his progressive base.
The chaos surrounding the caucus process may limit the impact of the Iowa outcome: It is already apparent that, in a departure from past presidential campaigns, no candidate intends to drop out as a result of a disappointing finish in Iowa. Several candidates, including Mr. Biden and Ms. Warren, emphasized on the night of the caucuses that the nomination would be decided over a long process.
And by late afternoon on Tuesday, every major candidate — and a number of lower-profile competitors — had already moved on to New Hampshire, the next state in line, which holds a primary on Feb. 11.
Arriving there on Tuesday afternoon, Mr. Sanders struck a balance between lamenting the procedural breakdowns in Iowa and defending the ultimate significance of the vote there. The caucuses, he acknowledged, were “not a good night for democracy.”