Atletico had trailed hosts Bilbao 1-0 at half-time but were inspired by Griezmann after the break - the French forward dragging the visitors level in the 46th minute before going on to complete his first hat-trick for the capital club.
Simeone claimed Griezmann's improvement in the second half at San Mames mirrored that of the rest of the team with Atletico's coach declaring it was the type of performance that had encouraged him to sign the France international in the first place.
"In the first half he [Griezmann] couldn't find the way to get into the game and be able to break in terms of his speed," Simeone said at his post-match media conference.
"In the second half, it was the opposite. He made a terrific second half.
"The player we needed, the one we went looking for. We trust his characteristics very much and we need him."
The win sees Atletico enter La Liga's winter break in third on 35 points, three behind second-placed Barcelona, while Real Madrid, who have a game in hand, lead with 39.
Simeone was thrilled to complete 2014 - the year they won their first league title in 18 years - with a win.
"The first half was hard for us. It was not good but bad," the Argentine coach said.
"We could not find our place, we were undefined because of them. We didn't win the cleared balls.
"The second half was much better. The team felt calmer in their positions and managed to close 2014 as a huge year.
"[It was] great because of a group of players who move forward no matter how big the difficulty of the situation. Today wasn't an easy match, losing 1-0 and playing poorly, and they responded in a wonderful stadium with fans who always respond and with an extraordinary second half.
"I'll say it again, congratulations because it's not easy to come to this field and to win."
With Mario Mandzukic suspended, Griezmann started up front for Atletico, flanked by Arda Turan and Saul, but the 23-year-old was barely sighted in the first half, as Mikel Rico gave Bilbao the lead in the 17th minute.
But in the second period, Griezmann converted a header on Atletico's first attack and then scored on the break in the 73rd and 81st minutes to cap off the champions' triumph.
"I played my best game since coming here and I have to continue on this path," the former Real Sociedad winger said.
