Uruguay centre back Gimenez dithered close to the byline and Lucas Perez robbed him of the ball and charged towards goal before coolly slotting past goalkeeper Jan Oblak 13 minutes from time to level the scores after Tiago had put Atletico in front before halftime.
Atletico, who travel to Kazakhstan to play Astana in the Champions League on Tuesday, are third on 20 points from 10 matches, with Real and Barca on 21 ahead of Saturday's games at home to promoted Las Palmas and away to Getafe respectively.
Some of Atletico’s new signings, like Jackson Martinez and Yannick Carrasco, have been impressive in recent weeks but it was midfield stalwart Tiago who fired them ahead in the 34th minute in Galicia.
A loose ball fell to the Portugal international just outside the penalty area and he cracked it into the corner of the net past Deportivo keeper German Lux.
Atletico defender Diego Godin headed against a post at a corner five minutes later and Antoine Griezmann poked the rebound straight into Lux's hands.
The visitors seemed content to soak up some ineffective Deportivo pressure in the second half without going for the killer blow and were punished for their lack of ambition when Perez struck the equaliser.
Deportivo, who are sixth ahead of the weekend's games, came close to taking all three points when midfielder Faycal Fajr curled a long-range effort on to the corner of the goal frame two minutes from time.
"It was an unfortunate passage of play," Atletico midfielder Koke told Spanish television.
"But we have to carry on and on Tuesday we have a Champions League game we must win," added the Spain international.
"We come up against some good first division sides, who have players of quality, and they can beat you or end up drawing which is what happened today."
Celta Vigo, who play at Real Sociedad on Saturday, have 18 points in fourth, one ahead of fifth-placed Villarreal before their game at home to Sevilla, who are eighth, also on Saturday.
(Corrects spelling of Gimenez)
(Reporting by Iain Rogers, editing by Mark Meadows)
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