Atletico coach Diego Simeone picked a strong side on Thursday in contrast to his opposite number Jose Luis Mendilibar, who made six changes from his last league outing, and watched his players dominate the Basque minnows in a one-sided affair.
Top scorer Griezmann headed Atletico in front in the 28th minute after Stefan Savic had an earlier goal ruled out for offside, while Argentine winger Correa doubled their advantage in the 60th by sweeping the ball in after a run by Yannick Carrasco.
Substitute Gameiro stretched Atletico's lead further by heading in from two metres out in the 67th minute after replacing Correa to give Eibar little chance of turning the tie around in the second leg at their Ipurua stadium next Thursday.
Holders Barcelona visit Real Sociedad in the final quarter-final first leg later on Thursday. Real Madrid were beaten 2-1 at home by Celta Vigo in their first leg game on Wednesday, while Alaves beat second division outfit Alcorcon 2-0.