Kalla was more than 20 seconds behind Finland and Germany when she started her first stretch, but quickly caught up.
In the home stretch, Kalla got past Finnish Krista Lähteenmäki and German Nystad in a hellish spurt.
Norway was favourite to win, but ended fifth. Finland ended second and Germany third. The win marked Sweden's first victory in the women's 4x5-kilometer relay since 1960.
"I tried to gather as much pace as possible in the final curve and it was fantastic to be the first to cross the finishing line," Kalla told Swedish Radio's sports department.
Kalla has now won three medals in total in Sochi, after taking home the silvers in the skiathlon and in the 10-kilomentre classic.
This means Sweden has got eight Olympic medals in total so far: one gold, five silvers and two bronzes.