DES MOINES, Iowa—Hillary Clinton is making inroads with Iowa Democrats who had qualms about her candidacy, after turning in a solid performance at the party’s first debate and avoiding missteps in a marathon appearance before a Republican-led committee examining a terrorist attack in Benghazi, Libya, polling and interviews show.

With Vice President Joe Biden announcing last week he would stay out of the race, many Iowa Democratic leaders and rank-and-file voters said Sunday Mrs. Clinton had reclaimed a commanding position...