He added that winning Europe's second-tier competition represented "more than a trophy - it's a victory for everything we stand for, for hard work and industry".
Valere Germain's frustrating miss from six yards after Dimitri Payet had played him through on goal was Marseille's only clear-cut chance in the first half.
The irrepressible Griezmann started and finished the move for his second, four minutes into the second half.
Atletico will face French club Marseille in the final to be held at the Groupama Stadium in Lyon, with kick-off set for 20:45 SA time.
Atletico's triumph comes after agonising defeats to Real Madrid in the Champions League finals of 2014 and 2016.
"But we have experienced players too, players who have played in and won European finals, the Champions League in the case of Luiz Gustavo". Marseille have not been in a European final since losing the 2004 UEFA Cup to Valencia, and Simeone said his side's superior experience in finals would help his side but would not be decisive.More news: Flags half-staff for 'Peace Officers Memorial Day'
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"The truth is this was a dream for me, to win a trophy with Atletico, with [Diego Simeone] and my friends and teammates". They will receive €10.3 million in total for their run in the Europa League. Subs: Juanfran (on for Vrsaljko, 45) 7, Thomas Partey (on for Correa, 88) N/A, Fernando Torres (on for Griezmann, 90) N/A.
"He leaves a legacy of never surrendering, never lowering his guard", Simeone said.
The football gods demonstrated it was not to be their night as a Kostas Mitroglou header bounced off the inside of the post and out before veteran Gabi had the final say after another assist from Koke.
Atletico's next challenge will be trying to stop a bigger club signing Griezmann, amid ongoing interest from Barcelona.
Griezmann said: "It is not the moment to talk about my future, I must enjoy having won a trophy, and celebrate it with the fans".
But before we all go ahead and start saying how Griezmann is hilariously original with doing these celebrations, the first group I remember seeing do this is the UMBC basketball team.