Barcelona's 3-0 win over Celta Vigo in the night's earlier kickoff saw Atletico move down to second, but a narrow victory at the Vicente Calderon ensures that Los Colchoneros climb back to the summit.
Granada began growing in confidence as the half passed by, and Youssef El-Arabi almost gave the Nazaries a shock lead after 32 minutes, getting on the end of a Fran Rico set piece but failing to convert the chance.
Atletico thought they had taken the lead shortly after courtesy of Tiago, though his 'goal' was ruled out after the referee ruled that he had pulled back on an opposition player in the build-up.
Diego Simeone's side finally made the breakthrough shortly after the hour, with Costa converting the Red and Whites' 10th corner of the evening, making no mistake heading past Roberto to fire his team ahead.
The hosts went in search of a second and came close to finding it through Gabi, but his long-range half-volley was kept out once more by Roberto to keep his side in the contest.
As the game edged towards a close it was Atletico who looked more likely to find the game's second, but the match eventually petered out ensuring all three points for the home side.