Costa headed Jose Sosa's cross past Granada keeper Roberto to ensure Atletico moved one point clear of Barcelona, who slipped to second, and three clear of capital city rivals Real Madrid, who lost 2-1 at Sevilla.
The Vicente Calderon faithful were left to sweat for the breakthrough against their mid-table opponents, though, with the Spain striker scoring in the 63rd minute.
The result in Madrid, and the shock final score in Sevilla, was almost the perfect outcome for Atletico and manager Diego Simeone, who assumed narrow in control of the title race with eight league games remaining.
Barcelona had earlier won their match against Celta Vigo 3-0 to keep the battle very much still on the boil.
Joao Miranda had a chance to tip the game in Atletico's favour early on only to fire over the crossbar before Arda Turan sent his shot wide.
Despite their best efforts, Atletico were struggling to get a shot on target in the first half, although luckily for them Granada were not faring any better.
After the break, Granada defender Jeison Murillo put a close-range header wide before David Villa forced a save out of Roberto at the other end with a header from slightly longer range.
Costa had a good chance to break the deadlock on the hour mark with a header from the six-yard box, but Roberto made the save.
However, the 25-year-old was not to be denied three minutes later when he got onto the end of Sosa's cross to head home from a similar range.
Atletico midfield Gabi tried his luck with a 30-yard blast with 15 minutes remaining, but Roberto made the save, and that turned out to be the last meaningful chance for both sides.