Griezmann scored his 100th goal for the team to give them victory.
A shot was blocked on the line and a late headed goal was saved for Athletic.
The hosts were not able to hold onto third place.
Jan Oblak was forced to leave the game at the end of the first half after he suffered a heavy fall.
Athletic threw everything at his replacement, Ivo Grbic, who was forced into a full stretch diving save to keep out the headed goal.
The hosts had a penalty ruled out by the VAR at the start of the second half.
A battling Atleti side had six players booked in total, including three of their subs, but held on for a sixth league win of the season.
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