In the men's event, five Norwegians made it through the early stages and to the final and but it was Klaebo finished first.
The 23-year-old crossed the line in 2:52.07 with compatriot Paal Golberg 1.2 seconds behind and fellow Norwegian Erik Valnes rounding off the podium, finishing 1.6s behind Klaebo.
There was another home victory in the women's event as Falla edged a close race, getting the better of Sweden's Jonna Sundling by 0.48s.
Meanwhile, Switzerland's Nadine Faehndrich finished 2.88s back to claim bronze and her first sprint World Cup podium.
Klaebo leads home Norway 1-2-3-4