“I’m so proud to be a Democrat tonight,” Mr. Insalaco said, “and so proud that we can call this woman one of our own.”

Mrs. Clinton’s primary rival, Senator Bernie Sanders of Vermont, played a symbolic role in hopes of unifying the party behind her. After Vermont arranged to go last in the roll call, Mr. Sanders joined its delegation to roars of “Bernie, Bernie” and called on the party to rally behind Mrs. Clinton.

But it was the appearance of Mr. Clinton, shortly after 10 p.m., that stirred the crowd most, as he set out to share a more personal side of the sometimes-reserved former secretary of state.

Unspooling memories of their 45 years together, Mr. Clinton used warm and detailed anecdotes to argue that the couple’s political enemies had spent decades creating a “cartoon” of his wife that he was now determined to puncture. Mrs. Clinton is among the most unpopular presidential nominees in modern history, and the former president appealed to the audience to see through the political attacks on her.