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It was a sign of what was to come in the first half as the hosts were forced back without causing much trouble at the opposite end, but Atletico lacked a cutting edge until finally making a breakthrough 11 minutes before the interval.
A patient move from Atletico ended with the ball reaching Tiago, and the midfielder capitalised on space to find the bottom corner from the edge of the box.
Forced to take more risks in the second half, Deportivo tested Jan Oblak for the first time as Federico Cartabia ended a decent spell by forcing the goalkeeper into a save from close to 25 yards.
Deportivo continued to commit more men to their attacks as Atletico protected a slender advantage impressively after conceding just twice in their previous six league games.
However, a moment of hesitancy offered the equaliser to Deportivo as Lucas dispossessed Jose Gimenez before showing impressive composure to find the bottom corner with a fine finish.
A once cagey affair became open late on as Atletico went in search of a winner, but it was Deportivo who came closest to finding the third goal when Faycal Fajr's effort crashed against the crossbar in a dramatic finish.