The combative striker, who has been called up to Luiz Felipe Scolari's Brazil squad for their friendlies against Italy and Russia, finished off a counter-attack at the second attempt in the 35th minute.
Just after the re-start he guided Koke's free kick into the net from close range and he came close to completing a hat-trick in a typically bruising encounter in Pamplona.
Costa was forced to limp off with an ankle injury midway through the second half.
After two games without a win or a goal, Atletico moved on to 60 points in third with 10 games left to play, one behind their Copa del Rey final rivals Real, who came from behind to beat Real Mallorca 5-2 at home on Saturday.
Leaders Barcelona, with 71 points, were playing Rayo Vallecano at the Nou Camp in the late game.
Atlético Madrid had 40.6% of possession.
Atlético Madrid had five shots, including seven on target.
Cristian Rodríguez made a big difference on the match for Atlético Madrid after coming on for Arda Turan in the 86th minute, as he took two shots.
Falcao had four shots for Atlético Madrid, more than any of his team-mates.
Thibaut Courtois pitched in with five stops.Osasuna managed to dominate the possession game, and maintained control of the ball for 59.4% of the match.
Osasuna managed 19 shots, but it was not enough.
They also completed 318 passes and had 36 crosses in the match.
The most effective of Rojillos's three substitutions was Álvaro Cejudo, who, after replacing Francisco Puñal in the 73rd minute, took two shots.
The seven attempts by Oier were the most of any Osasuna player.