Falla won by .38 of a second, a comfortable margin in the world of sprint skiing, over silver medalist Ingvild Flugstad Oestberg of Norway.
The bronze went to Vasna Fabjan of Slovenia.
American Sophie Caldwell reached the finals, but fell early in the race and finished sixth. It was the highest finish ever in an Olympic cross-country race by American woman.
2014: NFL Cheerleaders [PHOTOS]
Michael Sam cut from the Dallas Cowboys