It was the third medal of the Sochi Games for Kalla, who anchored a team that included Anna Haag, Ida Ingemarsdotter and Emma Wiken.
Finland won the silver and Germany took the bronze, both finishing less than a second behind. Norway, defending champion and traditionally a strong force in the 4x5-kilometer relay, wound up fifth. It was only the second time in the last 10 Olympics Norway had failed to medal in the race.
The women's relay has been contested in the Olympics since the 1976 Games in Innsbruck and this was Sweden's first medal of any kind in the race.
Saturday's event was essentially a three-country race from the start, but when Haag dropped back during the third leg it appeared only Finland and Germany would be left to fight for the gold.
Finland's Krista Lahteenmaki started out on the final leg with a five-second lead over Denise Herrmann of Germany.
Kalla, silver medalist in both the skiathlon and 10-kilometer classic race, began 25.7 seconds behind.
It takes a world-class female cross-country skier about 14 minutes to cover five kilometers, so that was how much time Kalla had to make up the large deficit. She steadily moved closer to the leaders, however, fighting her way through the soft snow on another 50-degree day in the mountains.
Kalla eventually passed Lahteenmaki and Herrmann in the home stretch in front of the grandstands and brought her team a victory in 53 minutes, 2.7 seconds.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys