Granada started brightly and almost took the lead after just 13 minutes when Yacine Brahimi found space in the box, but the striker saw his header come back off the crossbar.
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Then, with a few minutes to go until the break, Diego Mainz bundled Villa over inside the box and the referee awarded a penalty. Costa made no mistake, slotting the ball into the corner for his 12th goal of the season.
The visitors were comfortable in the second half but didn't really come close to scoring until 20 minutes from the end, when only a wonderful save denied Costa his second of the game after being found inside the box by Villa.
Atletico did double their lead shortly after though, again from the penalty spot. Villa was the man fouled once more, and this time he took it himself, calmly stroking the ball home.
There was drama at the end of the game though, as Felipe Luis received a second yellow for handling the ball, reducing Atletico to ten men before a late goal from Odion Ighalo made for a nervy finish.