Colombia forward Falcao was sold to Monaco for more than 50 million euros ($66.66 million) in May after scoring 28 league goals for Atletico last season.
Costa had reportedly also been a multi-million euro target for Liverpool, but last season's third-placed finishers hung on to the powerful frontman and last week extended his contract to 2018.
He returned the faith shown in him with a poacher's goal on Sunday, volleying in the opener at a corner in the 35th minute and, after Sevilla had equalised, tore onto a through ball to strike the second low into the corner 11 minutes from time.
Cristian Rodriguez scored a wonderful solo effort to make sure of the points for the visitors at the end.
"Costa had a very good pre-season, putting in a lot of work, and he knows he has to be physically strong at all times," coach Diego Simeone told reporters after a game that finished at almost one in the morning local time.
"He is very important for us because of his character which is contagious for the rest of the squad.
"Goals are a consequence of all this. Diego unsettles rivals and opens up the pitch for us."
Costa, a player very much cut in the same abrasive image as former Argentina captain Simeone, was the ideal foil for Falcao last season with his physical presence and uncanny ability to get under the skin of his opponents.
He had to be restrained by his colleagues from getting involved in a dispute with Sevilla players as the teams left the pitch at halftime.
His new strike partner is Spain international David Villa, signed from Barcelona for a fee that could rise to 5.1 million euros, but the 31-year-old had a relatively quiet competitive debut and was substituted in the second half.
"Falcao will be missed because he is a great player but we have Villa who is different type of beast," Costa told reporters.
"We know he will do just as well as Falcao. His work and commitment to the team is impressive. He is humble and has a great attitude. I think his quality shows and he will give a lot to Atletico."
Villa will get to meet his former team mates on Wednesday when Atletico host the La Liga champions at the Calderon in the Spanish Super Cup first leg.
($1 = 0.7500 euros)