Iker Muniain's delightful 37th-minute opener was cancelled out 129 seconds later by Diego Costa, who claimed his third goal of the season and first since October.
The striker paid tribute in his celebration to Atletico's women's team midfielder Virginia Torrecilla, who underwent successful surgery to have a brain tumour removed last month. Costa showed a shirt bearing her name.
Substitute Santiago Arias came closest to snatching all the points for Los Colchoneros, who sit sixth in the table on goal difference.
Marcos Llorente opened up Athletic with a fine diagonal throughball in the 12th minute, sending Yannick Carrasco scampering through to shoot just wide across goalkeeper Unai Simon, who briefly lost his footing.
Jan Oblak was uncharacteristically wobbly when he took two attempts to grasp Inaki Williams' 25th-minute header but the Slovenia international displayed his brilliant best when he kept out Yeray's glancing attempt at full stretch.
Oblak had no chance when Muniain dispatched a clever finish with the outside of his right boot into the bottom right corner.
Atletico sprung immediately from their mid-half slumber – Koke's cute pass leaving Costa to slide a finish beyond Simon.
That seemed to inspire a role reversal during the early stages of the second half, with Atletico enjoying plenty of attacking territory but Athletic keeping their visitors honest on the break as Yuri Berchiche crashed an effort into the side-netting.
Diego Simeone's anguish in the dugout was clear when Arias arrived unmarked at the back post 10 minutes from time, drawing a brilliant point-blank save from Simon.
What does it mean? Atletico away form still a worry
Simeone's side might have memorably prevailed at one of European football's great fortresses in their previous competitive match, but that victory at Anfield cannot obscure the fact that they have now won just one of their past 11 away games in LaLiga. They are level on points with Real Sociedad and Getafe in fourth and fifth respectively and such diminishing returns on the road are why their battle for Champions League football will probably go down to the wire.
Muniain and Garcia roll back the years
Lockdown has done nothing for anyone's conception of time and dates, but when Athletic's creative forwards took control of the first half, it was easy to imagine being somewhere during the early stages of the previous decade. Muniain – still only 27 – was irresistible at times, evoking memories of when he was a wonderkid and Spanish football's next superstar in waiting. Playing against the club he starred for in their 2013-14 title win, Raul Garcia proved a superb foil for the number 10.
Llorente ill-suited to number 10 duties
Marcos Llorente's two-goal heroics against Liverpool undoubtedly played a part in Simeone's decision to start him in an advanced role behind Costa. Although the Spain international's play was typically neat and he did not lack for work rate, the absence of his creative qualities from deep as the likes of Thomas Lemar and Alvaro Morata awaited second-half introductions was perplexing.
What's next?
Atletico are on the road again at Osasuna on Wednesday, when Athletic travel to struggling Eibar.
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 LaLiga, with Sunday's behind-closed-doors encounter at San Mames their 13th draw of the campaign – setting an unwanted club record.
"I believe it was a very strength-based game. It was a tough situation to be in," Koke told LaLiga TV, after the draw moved sixth-placed Atleti level on points with Getafe and Real Sociedad, the latter of whom take on Osasuna in Sunday'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."
Copa del Rey finalists Athletic lie 10th, five points behind Valencia in seventh, which is likely to be the last Europa League qualifying spot should they lose to rivals Sociedad in the showpiece.
Throughout his eight-and-a-half-year tenure, Simeone has expanded Spain's big two to a big three – most notably when Atleti pipped Barcelona and Real Madrid to the title in 2013-14.
However, they lie sixth in the standings after Saturday's stalemate at San Mames, with a 13th top-flight draw this season setting an unwanted club record.
In their previous competitive match before the coronavirus shutdown, Atletico sensationally dumped holders Liverpool out of the Champions League to reach the quarter-finals.
However, their participation in next season's competition is far from assured and fourth-placed Real Sociedad could open up a three-point advantage when they host Osasuna later on Sunday.
"The aim for the two teams was to always try to win, we started well in the first 10 minutes, then they positioned themselves better and they won the second ball and they did not let us attack," Simeone told reporters, after watching Diego Costa find a quickfire response to Iker Muniain's opener.
"The goal came, the reaction, after a very well-constructed play.
"In the second half we were closer to that victory. Santiago Arias' chance was the clearest. He could have allow us to take the victory in a stadium where you always have to work to win.
"We have one final left in each match."
Simeone picked two-goal Anfield hero Marcos Llorente in an advanced role behind Costa, leaving striker Alvaro Morata on the bench until the 63rd minute.
Nevertheless, the Atleti boss would not rule out pairing them as a front two in future, with a Wednesday trip to Osasuna in LaLiga on the agenda.
"Seeing what I see and knowing the squad, I interpreted this situation to be like this today," he said.
"Over time, they will be able to coexist. We understood that Marcos could give us some incision like he did, with some sprints, some passes and the dirty work."
