Costa turned Juanfran's low cross into the net in the 69th minute for Atletico's third goal in the first leg of the last 16 tie against third division Lleida but immediately limped to the sidelines. He did return to the pitch, however, and completed the game.
Diego Godin had headed Atletico in front in the 33rd minute following a free kick and Fernando Torres slid in to double the lead, completing a sweeping counter-attack although he was fortunate not to be ruled offside.
Antoine Griezmann completed the rout with a goal from a free kick in stoppage time, moments after the home side had defender Marc Trilles sent off for a second yellow card.
The win put Atletico on the verge of the quarter-finals, with the second leg at their Wanda Metropolitano stadium on Jan. 9.
Spain international Vitolo, a 40 million-euro signing from Sevilla who spent the first half of the season on loan at Las Palmas, also made his debut for Atletico by coming on in the second half.
Costa completed a 57 million-pound move back to Atletico from Chelsea in September, having left the club in 2014 after firing them to the La Liga title. The move was delayed by the club's ban on registering new players which expired on Jan. 1.
Costa his side an extra edge, charging into challenges and provoking opponents as well as starting and finishing the move for the third goal, his first strike for Atletico since scoring against Chelsea in April 2014.
Last year's runners up Deportivo Alaves won 3-1 at tiny island side SD Formentera. Valencia visit Las Palmas and Sevilla visit Cadiz later on Wednesday.
(Reporting by Richard Martin, editing by Ed Osmond)
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