Julian Alvarez's second-half spot kick cancelled out Viktor Gyokeres's first-half goal for Arsenal, leaving Atletico arguably the more frustrated side after a match they largely controlled but could not fully bend to their will.
The hosts had chances to take a lead into Tuesday's second leg but lacked the final touch, and now face the hefty task of beating Arsenal at the Emirates to reach the final.
"Our second-half performance was much better in terms of intensity and pressing. That's the way to stay in the tie. It's going to be a great match to play. Whether you're at home or at the ground, it'll be a nail-biter but that's the beauty of football," forward Griezmann told reporters.
"Conceding at the end of the first half hurts. We made two or three tactical adjustments to press better and shift our shape, and that made the difference. Then our fans really pushed us on."
Griezmann, who had a chance for the winner but struck the crossbar in the 63rd minute, said Atletico had to use their improved second-half display as the template for the return.
"I had three clear chances to score late in the second half but I couldn't find the net; I'll get one in the second leg for sure. We all did a great job and we have to take the second half with us," Griezmann said.
"My big dream is to reach the final. I've prepared a lot for this match and will be ready for the second leg, hopefully we can achieve it."