Bjoergen, who won the individual sprint in Vancouver four years ago, was seeking her fifth Olympic title but failed to make the final where Falla was never threatened.
Team mate Ingvild Flugstad Oestberg produced a late surge to catch Slovenian Vesna Fabjan on the line and take silver by two hundredths of a second.
"It's just unbelievable. I can't believe what I just did. I've been dreaming of this since I was 3-years-old," Falla told reporters.
"It's been a hard time for the Norwegian team after what happened with Astrid. Astrid is really strong and she's taken the decision to stay here in the Olympics."
Astrid Uhrenholdt Jacobsen, whose brother died suddenly last Friday, reached the final as one of two "lucky losers" but could only manage fourth place.
However, there was more success for the country later on Tuesday when Ola Vigen Hattestad took the men's sprint gold.
Bjoergen, who won the skiathlon on Saturday, had taken third place in the qualification round as several skiers fell on the 1.3km circuit on melting snow in mild conditions at the Laura Biathlon and Cross-Country Complex.
"I was not good enough. I had to be faster but I'm really happy, my team was incredible today, we got two medals," Bjoergen told reporters.
"It's a difficult track, but it's the same of everyone, and the best on it is the winner."
German Denise Hermann and American Kikkan Randall, first and second in the sprint World Cup standings, were eliminated in the semi and quarter-finals respectively.
Poland's Justyna Kowalczyk, second behind Bjoergen at the Vancouver Olympics, is sitting the event out as she nurses a micro fracture in her foot to be ready for the remaining disciplines.
Oestberg and Fabjan seized their chance after these upsets.
"It means everything. It's been a goal for sure this year, but also for four years now. It's unbelievable that I got a medal, and silver - that's like gold for me," said Oestberg.
Fabjan was equally pleased.
"It's everything," she said. "It's 20 years of training. My dreams have come true.
"I wanted this medal so much. In the last curve I was still in second. I said to myself this should be my best finish ever and in the end, I got a medal."