Mario Bermejo gave Celta an eighth-minute lead only for Juan Dominguez to equalise 21 minutes later in their La Liga clash.
Celta were then reduced to 10 men in the 52nd minute when Gustavo Cabral was dismissed following his second yellow card but the visitors failed to make the extra man count as they claimed their third away point of the season.
The hosts took an early lead when Iago Aspas broke the offside trap and ran into the area.
Visiting goalkeeper Daniel Aranzubia could only parry Aspas' rather scuffed shot with Bermejo pouncing on the loose ball to slot home into the bottom right-hand corner of the goal from close range.
There was little immediate reaction from Deportivo, rather more attacking play from the hosts with Aspas and Cabral both shooting wide.
Ivan Riki curled a free-kick over the bar with the first shot for the visitors but after showing little as an attacking threat, Deportivo drew level just before the half hour.
Juan Carlos Valeron passed into the path of Dominguez and he slotted home a right-footed shot into the bottom right-hand corner of the goal.
Alex Lopez's header from a Michael Krohn-Dehli corner was comfortably taken by Aranzubia as a patchy first half ticked down.
The atmosphere intensified seven minutes into the second half when the hosts were reduced to 10 men following Cabral's dismissal, his foul on Riki bringing him a second yellow card.
There was no immediate effect, Celta continuing to attack. Aspas headed over while Roberto Lago's first-time shot along the ground curled wide.
The hosts fashioned an opening through neat interplay with Hugo Mallo's short pass finding Aspas whose left-footed shot curled past the left post with the keeper beaten.
Deportivo went down the other end, Bruno Gama sending in a dangerous cross from the right which found Nelson Oliveira but he directed his downward header wide.
Celta then had the ball in the net when Andreas Tunez headed home but he was a yard offside.
While there was plenty of effort from both sides, there was little to really test either goalkeeper.
However, in the third minute of injury time Celta momentarily held their breaths when Diaz Ayoze's fierce volley struck Mallo but the referee waved away appeals for handball.