Norway could do no better than fifth after winning all the World Cup and world championship relays since the Vancouver Games, where they also won gold.
Finland took silver and Germany won the bronze.
Kalla started the final leg 25.7 seconds behind Finland in third position but caught Krista Lahteenmaki and German Denise Herrmann some 300 metres out and won the sprint to the line.
Kalla is the first athlete at the Games to win three medals after claiming silver in the skiathlon and the 10km classic.
It is Sweden's first gold at the Sochi cross country and they now have the most medals (8) ahead of Norway (7), who have won two titles.
Norway were never in contention on Saturday and Marit Bjoergen, who has been struggling since taking the opening skiathlon event, could not match Kalla's furious pace.
"I feel like I was strong but there were so many people out there it was hard to overtake," Norway's Heidi Weng told reporters.
"I don't know why I didn't do too well. I'm not sure if it was the skis or something else."
(Additional reporting by Dmitry Rogovitskiy; Editing by Peter Rutherford and Robert Woodward)
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