Both Atletico and Barca have not dropped a point in six league games so far and the capital club were on course to their latest success when Costa struck twice in seven first-half minutes.
Oriol Riera kept them honest for the second half after he struck just before the break, but Osasuna never threatened an equaliser to crash to their fifth defeat of a forgettable start.
Osasuna sounded an early warning in the very first minutes when Riera headed an Emiliano Armenteros free-kick over the crossbar.
Sisi then hit a post with a 18-yard strike and while Atletico had made na sluggish start they were soon 2-0 up thanks to their Brazil forward Costa.
The 24-year-old swept home the opening goal from near the penalty spot on 18 minutes, before doubling his side's advantage with a more memorable effort.
Costa latched onto Jorge Koke's right-wing cross before superbly heading home.
In between Costa's goals Arda Turan had found time to head narrowly wide and while Atletico had now begun to move through the gears their progress was slowed by Riera's strike on 43 minutes.
Mario Suarez fouled Abella Damia and from the resulting free-kick Armenteros picked out Riera to score from six yards.
Costa was denied his hat-trick shortly after half-time when Osasuna goalkeeper Andres Fernandez parried his shot while Armenteros could only muster a weak effort at the other end.
Damia wasted Osasuna's best chance when he shot over from close range and from there Atletico could have grabbed a third with Turan missing the target and Emiliano Insua twice going close.