The strikers bagged a penalty in either half before Odion Ighalo netted a late consolation, ensuring Diego Simeone's side moved back to within a point of Liga leaders Barcelona
The strikers scored from 12 yards in each half to secure maximum points for the visitors despite Ighalo's late consolation, with the hosts left to rue their profligacy in front of goal as they suffered a second consecutive home defeat in La Liga.
No team in Spain's top flight has scored fewer goals than Granada, but they began the match by far the more threatening of the two sides, with Manuel Iturra in particular causing problems for Diego Godin and Co.
Once again though, Granada couldn't turn possession into goals, and late on in the second half it was a clumsy challenge from captain Diego Mainz on Villa which gave Costa the chance to open the scoring, and the striker - who this week affirmed his desire to play for Spain - duly buried the spot-kick.
From that point, Atletico seized control of the game and mid-way through the second half made their pressure tell. A swift counterattack led to another trip on Villa, and the veteran striker made no mistake as he converted the penalty.
Granada mounted a late assault on goal after Filipe Luis was sent off for a second yellow card, and Odion Ighalo turned home a cross as the clock ticked over into stoppage time, but ultimately to no avail.
Tempers flared after the final whistle as Costa tangled with Iturra as the teams departed the pitch, but Atletico's 10th win in 11 games is all that will matter to Diego Simeone, as his side moved back to within one point of Liga leaders Barcelona.