'I am very happy for having once again felt like a footballer and for having played with Atletico Madrid again, ' Costa said following the victory.
Costa appeared to have injured himself stretching to score the goal, but carried on and finished the game.
Brazil-born Costa joined Chelsea from Atletico for a reported £32m in 2014 and he scored 58 goals in 120 appearances for the Blues, including 20 in the Premier League last season as they won the title.
Diego Simeone will leave Atletico Madrid in the future and take up the reins of Internazionale - according to his own son Giovanni. But he couldn't be registered as a La Liga player again until this January transfer window, which marks the end of Atleti's transfer ban stemming from a Federation Internationale de Football Association investigation into an under-18 player who previously appeared for the club.
"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".
'It is great for me to play for Atletico Madrid again, the club are very important to me and I am very happy with this.
The second leg against third-tier Lleida Esportiu is on Tuesday, with Atletico hosting Getafe in the league on Saturday.
Antonio Conte told Costa by text message that he was not in his plans at the end of last season.