The home side started well but tired in the second half and were lucky to escape with a point in the end.
Espanyol's Kiko Casilla rushed out of his goal to punch the ball away but it dropped into the path of David Zurutuza, who opened up his body and volleyed it over the defence and into the empty goal from 30 yards out.
After the break Sociedad started to dominate. Griezmann and his strike-partner Imanol Agirretxe were both guilty of missing the target from good positions.
Sociedad continued to keep Casilla busy throughout the half and 15 minutes from the end, their pressure told, albeit in unfortunate circumstances.
A chipped ball over the top found Griezmann, whose first touch bounced against Espanyol defender Javi Lopez and past the goalkeeper.
The visitors continued to push forward for an equaliser, with Griezmann missing yet another good chance, but were ultimately disappointed and had to settle for a point.