Diego Godin scored the only goal of the game five minutes before the break with a scrappy effort that found the net.
The visitors, who have a dreadful record this season on the road with only one victory, began well and should have been ahead after a quarter of an hour but Koke saw his volley well saved by Osasuna keeper Andres Fernandez.
Top scorer Falcao then went close for Diego Simeone's men but could not find the target.
The Colombian forward played a vital role in the goal moments later by forcing a save from Fernandez, with the resulting loose ball falling to Godin, who bundled home.
Falcao then wasted the best chance of the match by rounding Fernandez expertly before miscuing his shot.
The home side finally began to impose and laid siege to the Atletico goal as the fixture drew to a close.
Atletico are now just two points adrift of Levante and Espanyol, in fourth and fifth respectively, in the La Liga table.