Atletico players hail Europa performance
And the Brazilian playmaker, who was suspended when his Werder Bremen team reached the 2009 final, was ecstatic after his starring role this time around.
"It was a hard game but we played extraordinarily well," he told uefa.com. "We were almost perfect tonight even though we were facing a great team. It's a dream, an incredible moment."
For fellow goalscorer Falcao, whose brace took his season's tally in the competition to 12 and earned him the golden boot ahead of Schalke's Klaas Jan Huntelaar, victory was the crowning glory on a successful first season in Spain.
He said: "I came here to Atletico to win and I thank God for this opportunity tonight that very few other players can enjoy. I would like to thank my team-mates and the staff at this club and to my wife and family in Colombia; this was for them and for all Atletico fans."
Captain Antonio Lopez added: "The fans and players deserve this title. We put in a performance that was deserved of a final and we are so happy. We're now looking forward to joining our fans for the celebrations back in Madrid."
Athletic have delighted onlookers ever since they took advantage of Trabzonspor's elevation to the UEFA Champions League to come through a tricky qualifying round tie.
The zenith of their swashbuckling run to the final came with victory over Manchester United but tonight too many of their big names failed to rise to the occasion.
Coach Marcelo Bielsa said: "I feel tremendously disappointed. We didn't play well. Obviously, I'm responsible for how my team played and we did not accomplish our goal.
"Our opponents were deserving winners. There was a stark difference between the teams. We played how Atletico wanted us to, allowing them to play their way in the process."