Atletico's goal lived a charmed life in the final stages, thanks to brilliant defending from Reinildo Mandava and some impressive saves from Ivo Grbic, who had to come on for the injured Jan Oblak.
Ernesto Valverde's Athletic, now fourth, were awarded a late penalty for handball against Reinildo, but it was cancelled after the officials spotted the ball had struck the defender on the face.
Griezmann, who missed a penalty and various chances during two frustrating matches against Club Brugge in the Champions League, was delighted to net the winner.
"All the team did incredible work, that's us, all running together, attacking together, we it very well," said the forward.
"We did it against Club Brugge too but today with a bit of luck, the ball went in.