DES MOINES, Iowa—Former Secretary of State Hillary Clinton barely survived a surprisingly stiff challenge from Sen. Bernie Sanders in the Iowa caucuses, as the two candidates arrived in New Hampshire Tuesday morning to begin a weeklong dash to the nation’s first primary.

The Iowa Democratic Party declared the caucuses “historically close.” With all precincts reporting, the party’s tabulations put Mrs. Clinton’s delegate count at 49.8%, to Mr. Sanders’s 49.6%.

...