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Diego Costa's 39th-minute goal on the end of a clever Koke pass ensured the first-half advantage Iker Muniain's splendid strike granted Athletic was short-lived.
However, Atleti were unable to affect a full turnaround, with substitute Santiago Arias going closest to a winner when he drew a point-blank save from Unai Simon 10 minutes from time.
It means Los Colcheneros have now won only one of their past 11 away matches in La Liga, with Sunday's behind-closed-doors encounter at San Mames their 13th drawof the campaign – setting an unwanted club record.
Never before had Atletico been held to so many ties in a single season, and that number looks bound to grow even higher with 10 games still remaining in the current Primera campaign.
"I believe it was a very strength-based game. It was a tough situation to be in," Koke told LaLiga TV, after the drawmoved sixth-placed Atleti level on points with Getafe and Real Sociedad, the latter of whom take on Osasuna inSunday's late kick-off.
"We didn't take our chances. We have to suffer and accept the draw. We want to be in the Champions League and we need to get hold of the points.
"It's difficult from this position that we're in.
"They were better placed than us all over the pitch. We managed to change things after the first goal and a couple of minutes after found the equaliser.
"Santi had the chance at the end to get us the win."
Ex-Atleti forward Raul Garcia impressed for the hosts alongside the influential Muniain and rued his team-mates switching off as Costa hit back.
"We knew it was going to be a very difficult game," he told LaLiga TV. "There was a bit of speculation about how the teams were going to come back to the league action but we both went out to win the game.
"Once you've just scored you normally relax a little bit and that's what happened to us today.
"But we have to take the positives. We manged to compete against one of the best teams in the league and unfortunately we couldn't get three points."