Costa told reporters after a goal-scoring bow: "Now I am very happy because I feel like a footballer again".
Costa was farmed out to several clubs when he arrived at Atletico over a decade ago, before eventually pinning down a first-team place and landing a lucrative move to Chelsea in 2014.
The Brazil-born Spain global appeared to hurt his right knee in scoring, and received lengthy treatment on the sidelines, although he played on to see another sub, Antoine Griezmann, finished the scoring with a deflected free kick in added time.
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Costa has been in limbo since last summer when he was informed by Chelsea boss Antonio Conte via text message that he wasn't in the club's future plans.
A transfer ban meant that Costa could only be registered as a player for Atleti in January.
"I could not have asked for anything more on my second debut, to be back playing again and to have scored a goal is fantastic".
Simeone, however, is unhappy the 24-year-old has not been able to make the impact befitting his level of ability, with Vietto yet to score this term.
The second leg of the tie against third-tier club holds next Tuesday, with Atletico hosting Getafe in the league this Saturday.