Sergerie needed the Slovenian fighter to advance to the women's under-67 kilogram final in order to make one of two bronze medal matches. But Anic lost her semifinal match 7-0 to defending Olympic champion Kyung Seon Hwang of South Korea.
Anic, a taller competitor, landed three high head kicks to win 10-5 against Sergerie. The Canadian coaching staff challenged one of the kicks but lost when a video replay showed Anic making clear contact to Sergerie's face.
Earlier, Sergerie won her first round match. The 27-year-old from Sainte-Catherine, Que., defeated 17-year-old Farida Azizova of Azerbaijan.
The first point of that match did not come until the third round when Sergerie landed a kick to Azizova's body. The point held up when the judges ruled Azizova's counter high head kick as a miss after a video review.
Sergerie won the silver medal at the 2008 Beijing Games.
Hwang would go on to win ther second consecutive gold medal with a 12-5 result over Turkey's Nur Tartar. Just 26, Hwang also has a bronze in her collection, from the 2004 competition in Athens.
Helena Fromm of Germany and Paige McPherson of the United States earned bronze medals on Friday.
In the men's under-80 kilogram competition, Canada's Sebastien Michaud bowed out in the preliminary round to Armen Yeremyan of Armenia.
Michaud of Joliette, Que., was unable to hold on to his early 2-0 lead and lost 8-4.
He also had a chance in the repechage but was officially eliminated when Yeremyan lost his semifinal bout to Argentina's Sebastian Eduardo Crismanich.
Michaud finished ninth four years ago in Beijing.
Crismanich defeated Nicolas Garcia Hemme of Spain 1-0 in the final, with the bronze winners Mauro Sarmiento of Italy and Lutalo Muhammad of Britain.
Several of the division's top seeds made an early exit, including five-time world champion Steven Lopez of the U.S., world champion Yousef Karami of Iran and top-seeded Ramin Azizov of Azerbaijan.