Johannsson, 24, has signed a four-year contract and will be charged with the task of replacing Franco di Santo after the Argentine striker, last season's leading scorer at the Weserstadion, left for Bundesliga rivals Schalke 04.
Born in Alabama but of Icelandic origin, Johannsson has won 12 caps for the United States and scored nine times for AZ last season as they finished third in the Dutch Eredivisie.