The Brazilian was extremely doubtful to feature in the Group G encounter due to an ankle knock he sustained at Espanyol on Saturday but he ended the match as its star performer.
Raul Garcia opened the scoring inside eight minutes at the Franz Horr Stadium and Costa doubled the Spaniards' lead midway through the first half.
In the 53rd minute he struck his 12th goal in only 10 games to rubber-stamp a third win in three European fixtures for Atleti.
Simeone told his club's official website: "The truth is that he was a bit hurt after Saturday's game.
"The medical team and physios did a very good job and we expected him to be okay. The doctors gave him the all clear and he really put in a huge performance, working for the team.
"I really appreciate and value the work the guys are doing, it is great.
"They are putting in a lot of effort, a lot of commitment and a lot of teamwork and that allows us to be competitive and we're a tough team."
Costa, who missed previous victories over Zenit St Petersburg and Porto, had arrived in Vienna on crutches before hitting his brace and declared himself happy with his performance.
"I really wanted to play in the Champions League," he said. "It is a competition that every player wants to play in. I am very happy with my debut and very happy with the team's win.
"I had a bit of trouble and people knew but the medical department has worked hard so I could play."
The 25-year-old was born in Brazil and played in two friendly games for the Selecao in March, but recently declared his desire to play for Vicente del Bosque's Spain at the forthcoming World Cup finals.
The world champions are still waiting on final permission from FIFA - and Brazil - to call him into action but the player is in no doubt he has made the right decision.
"My mind is made up and people who need to know it already know," he added.