Costa scored the only goal late in the first half to give Atletico a 1-0 second-leg victory over Arsenal in the Europa League semifinals.
Arsenal's hopes of winning one last trophy under Arsene Wenger ended at the hands of Diego Costa and Diego Simeone's Atletico Madrid on Thursday.
Costa scored the only goal late in the first half to give Atletico a 1-0 second-leg victory over Arsenal in the Europa League semifinals, advancing 2-1 on aggregate to its fifth European final in nine seasons.
The result also meant Arsenal will not qualify for the Champions League for a second straight season.
"I'm very disappointed," Wenger said. "I'm frustrated as well because when you go out of the competition like we did, it's very difficult to take. I'm very, very sad to leave the club with this exit."
Wenger, in his 250th UEFA game as a club coach, announced two weeks ago he was standing down after nearly 22 years with the London team.
Atletico will meet Marseille in the final in Lyon on May 16 after the French team lost 2-1 to Salzburg following extra-time in Austria but went through 3-2 on aggregate.
Amadou Haidara and an own goal from Bouna Sarr put Salzburg 2-0 up on the night before Portuguese substitute Rolando struck in the 116th minute to send Marseille through.
Simeone's Atletico is already certain of a Champions League place next season because it is second in the Spanish League.
Atletico was demoted to the Europa League after failing to get past the group stage of the Champions League for the first time in five seasons.
"It was important to make it to the final," Costa said. "Atletico needed it, we are a great team."
Atletico will be playing its fourth European final under Simeone. It lost twice to city rival Real Madrid in the Champions League in 2014 and 2016 but won the Europa League in 2010 and 2012, the last time soon after Simeone had taken over.
"It's not easy to do what we're doing," Atletico midfielder Saul Niguez said. "We want to add the cherry on the cake at the end of the season."
Arsenal is sixth in the Premier League and needed to win the Europa League to qualify for Europe's leading club competition.
"We wanted to do it for the boss, he's had an amazing career," Arsenal defender Hector Bellerin said. "We wanted to try our best to finish at the top and that adds a little bit more disappointment for us."
Costa, the scourge of Arsenal several times during his three seasons with Chelsea, picked up a perfect pass by Antoine Griezmann and strode into the area before fending off a defender and lifting the ball over goalkeeper David Ospina.
Arsenal tried to rally but struggled to break through Atletico's stout defense, creating few scoring opportunities.
The hosts continued to threaten with Costa and Griezmann up front and had a few good chances to make it 2-0.
Both teams started cautiously at the Wanda Metropolitano Stadium, with Arsenal trying to control possession and Atletico playing typically tight at the back.
Arsenal lost captain Laurent Koscielny with a suspected Achilles injury less than 10 minutes into the match, putting in doubt his World Cup participation with France next month.
"It looks like an Achilles rupture," Wenger said. "But we have to be cautious. We will know more after a scan."
It was Atletico's 32nd game with a clean sheet in all competitions this season, the most of any team in the top five European leagues.
Atletico didn't have Simeone on the bench on Thursday because he was sent off in the first leg.
"History is made in these kind of games," said Simeone, who watched from the stands. "Tonight I felt the atmosphere in this stadium like a fan. They were with us every step of the way, they knew the side needed them."
