Atleti secured the win they needed to clinch a top-four finish in LaLiga on Saturday, despite playing over half an hour with 10 men following Mario Hermoso's straight red card for a foul on Loren Moron.
Costa was sent on as one of a trio of attacking changes just before Hermoso's dismissal and, after Diego Simeone's side absorbed some late pressure, he guided Yannick Carrasco's free-kick into the bottom-right corner to seal the three points.
8 - @atletienglish will play the European Cup for the eight season in a row, as many as in the previous seasons before Diego Simeone joined Atlético de Madrid. Installed#AtleticoMadrid pic.twitter.com/WQDyzKOIUf— OptaJose (@OptaJose) July 12, 2020
Betis defender Marc Bartra played Costa onside but a lengthy VAR check was conducted to see whether the Atleti striker had used his arm in the process of scoring.
The replays appeared inconclusive and the goal was allowed to stand, unlike first-half efforts for Angel Correa and Alvaro Morata that were chalked off with video assistance.
"It would be one thing if they called it offside for my goal but it didn't come off my arm. I'm very clear it wasn't [handball] and I'm happy," Costa told Movistar.
"It was a difficult game after Mario's red card. Our goal was to finish in the top four. Now we have that and we've got two more games we have to try to win to continue our good form.
"I think it was a good sign that we scored from a set-piece. We were in control. What happened in the second half made it more difficult, but we had a bit of luck and got the win.
"I don't know why we always have to suffer like this, but we always suffer and win. If we don't suffer, we don't feel right, so we're happy and we'll look for more glory."
Referee Jose Luis Gonzalez Gonzalez maintained his decision to dismiss Hermoso after a trip to the pitchside monitor, but Costa felt a booking would also have been a fair punishment.
"If they'd given a yellow card for Mario Hermoso then fine. I think it was on the limit and in the end the referee has to make the decision, but I think a yellow or red was fair," he said.
Guido Rodriguez disagreed with Costa's assessment of the goal, claiming the ball clearly struck the attacker's arm.
"We saw that when Costa scored it touched his arm, but they didn't make that decision, so that's how it was," the Betis midfielder said.
"We played with a player more, looking for the goal, but we leave pretty annoyed because we couldn't get it."