The 42-year-old Argentinian, who took up the Atletico reins in December, oversaw a 3-0 victory over La Liga rivals Athletic Bilbao in Bucharest as Los Rojiblancos picked up the trophy for the second time in three seasons.
Simeone told Spanish television channel Telecinco: "I'm happy for the players, the people, and because on Thursday the children can go to school with the Atletico shirt, and this is the most wonderful thing that can happen.
"I am very happy for Atletico Madrid. Very happy, very grateful to the players, my coaches, the physios, doctors. Were it not for them, the team could not respond in this way.
"We showed what Atletico are about - a fast team, strong defensively - and I'm very proud of this."
Comparing the success to his 1998 UEFA Cup win as an Inter Milan player, Simeone said: "Winning is always wonderful. As a player it is more enjoyable because you're on the field. You can scream, run around. As a coach you have to keep calm."
Radamel Falcao scored two sumptuous first-half goals for Atletico last night and Diego struck late on to seal a convincing triumph.
Athletic coach Marcelo Bielsa felt the scoreline flattered Simeone's side.
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"The margin of their victory was not deserved," he said. "The difference was exaggerated - it was not in proportion to what I saw.
"What happened on the pitch from my point of view is that they scored three goals and had eight chances; we had nine and scored none."
Athletic have delighted onlookers with their displays in reaching the final.
The zenith of their swashbuckling run came with victory over Manchester United but last night too many of their big names failed to rise to the occasion.
Bielsa added: "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."