Villa was withdrawn from the game four minutes before the break, having opened the scoring at the Estadio Vicente Calderon.
Following examinations, the club confirmed that the former Barcelona forward sustained damage to his left hamstring.
"David Villa suffered an elongation/strain without rupture (grade I) (of the) bicep femoris muscle on the back of the left thigh," the club stated on their official Twitter account.
Atletico have not set a timescale on Villa's return to action, although he will likely miss Wednesday's Copa del Rey semi-final first leg clash with Real Madrid at the Santiago Bernabeu.
Villa has 13 goals in all competitions since moving from the Spanish champions in July.
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