Atletico made a poor start to La Liga, resulting in the sacking of manager Abel Resino, but will finish the campaign in mid-table and with a European trophy.
They have also reached the final of the Copa del Rey and Sanchez Flores is delighted by the way the club's season has been transformed.
"I am really proud of my players. They deserve all the credit because they were able to overcome a complicated situation," he said.
"I can only stress my admiration for the work they have put in to turn our campaign around.
"It has been a long, hard season and that makes this success even more special.
"There are a lot of emotions that come out after an achievement like this.
"But I would like to dedicate this triumph to my four children and to Atletico fans everywhere."
It took extra time to separate Atletico and Fulham at the Hamburg Arena.
Diego Forlan was the star of the show, scoring both goals in the 2-1 victory with the winner arriving with five minutes remaining.
"We are thrilled to have won a very tough final and a very difficult competition," said Sanchez Flores.
"It is hugely satisfying - Atletico are a big, big club and we have made thousands of people happy.
"It's the players' moment, it is their time to celebrate. This is a massive occasion in the history of the club.
"This Europa League final was a big opportunity for our team, but it was never going to be easy and when you do manage to cross the line it is a fabulous feeling.
"Fulham were a very well-organised side and they made it very difficult for us to find that second goal.
"It was like we played two matches against that defence of theirs.
"They were great opponents but I would say that overall we deserved the victory."