Fontana makes Italian Olympic history in home Winter Games
The South Koreans won after taking advantage of a crash by the Dutch team 14 laps from the finish, although they only pulled ahead of Italy with one lap remaining.
Watched by Italian Prime Minister Giorgia Meloni, 35-year-old Fontana took the anchor leg and although she was unable to overhaul South Korea's Kim Gil-li, she cemented her place in Italian sporting history.
Canada finished in the bronze medal position.
Fontana also won a silver in the 500m and a gold in the mixed team relay at these Games and now has 14 Olympic medals, surpassing fencer Edoardo Mangiarotti, who won 13 medals between 1936 and 1956.
She has amassed three golds, six silvers, and five bronze medals over six Olympics.
Italy are enjoying their best Winter Olympics since Lillehammer in 1994. They sit second in the medals table behind Norway with nine golds and 26 medals overall.