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.
"I was inspired by the others in my team. I knew we had really good skis. When I saw the German I just wanted to go for gold," Kalla told a news conference.
"I would have been satisfied with bronze but I knew if I fought very hard it would be possible to cross the finish line first."
There was nothing Herrmann and Lahteenmaki could do.
"I tried to attack in the last bend but it was really hard. I thought I could make a good sprint but others were fast, too," said Herrmann.
"I did not think we could make the podium today so I'm still happy."
Lahteenmaki said she thought she was "going to collapse" and that she "had to fight to get that silver medal".
Kalla is the first athlete at the Games to win three medals after claiming silver in the skiathlon and the 10km classic.
Norway were never in contention on Saturday and Marit Bjoergen, who has been struggling since the opening skiathlon event, could not match Kalla's furious pace.
"It's hard to say what went wrong but I think we didn't have the best skis," Bjoergen told reporters.
In the 4x5km relay, athletes ski 2.5km in classic style before switching to freestyle skiing.
"We tried hard today but to win gold you need to have a good shape and good skis. We didn't have that today."
Team mate Therese Johaug had the same analysis.
"I think that we didn't have the best skis today and everybody saw that," she told reporters.
"But we did our best and the guys in the (waxing) cabin did their best too. We have to look forward to the next competition.
"We saw Sweden had good skis but that's part of the sport."
(Additional reporting by Dmitry Rogovitskiy; Editing by Peter Rutherford and Robert Woodward)
Share
