The La Liga champions were given a more stern test than they might have expected at against their third-tier opponents at Estadio Ramon de Carranza, but Garcia's strike ensured they came out on top.
Midfielder Garcia found the back of the net just before the hour mark with what appeared to be a cross, the ball looping into the far corner of the net following a neat passing move.
Cadiz posed a threat, but Diego Simeone's side were not to be denied victory as they geared up for the first leg of their Supercopa de Espana tie against city rivals Real Madrid next Tuesday.
Antoine Griezmann had an early chance to open the scoring, but the France striker headed wide when he was picked out by Gabi.
Fellow new boy Mario Mandzukic did manage to hit the target, but the Croatia striker's effort was kept out by Cadiz goalkeeper Oinatz Aulestia
Gabi fashioned another chance when he picked out Garcia with a cross, but the former Osasuna man headed off target.
Griezmann was in the thick of the action again when his fine strike 10 minutes into the second half brought another fine save out of Aulestia, then Mandzukic was replaced soon after being booked for an off-the-ball altercation.
Atletico got the breakthrough 59 minutes in, and only Garcia will know whether he meant to find the back of the net when he delivered the ball from the right and it looped inside the far post.
Cadiz gave a good account of themselves as they went in search of an equaliser but Atletico, who could have added to their tally, were able to record a win.
