Deportivo had Morocco midfielder Faycal Fajr sent off just before half-time following two quick bookings.
France striker Griezmann tapped home after good approach play from Kevin Gameiro with 20 minutes left, which was enough to see Diego Simeone's side close to within a point of Barcelona and two behind leaders Real Madrid.
"We couldn't play the game at the pace we wanted to," the Argentine said post-match. "We played at the pace that our opponents wanted us to.
"We found it difficult to play at the necessary speed and that began to make things complicated."