Striker Alvaro Morata got the visitors off to the perfect start by tapping an Angel Correa cross into an empty net in the opening minute but Celta's Fran Beltran levelled early in the second half on the volley.
Atleti are third in the standings on 63 points after 35 games and the draw edged them closer to securing a top-four finish as they moved nine above fifth-placed Villarreal, who are in action away to Getafe on Wednesday.
Celta meanwhile nudged a step closer towards guaranteeing survival in the top flight as they moved on to 36 points in 15th, seven above the relegation zone.
"It was a very evenly matched game," said goalscorer Morata.
"This is a very difficult ground to go to, Celta play very good football and don't deserve to be where they are in the table. They have a great squad and will get back towards the top in the future."
Once Beltran had equalised, neither side showed enough ambition to go and find a winner and the final whistle signalled a 15th draw of the season for each side, a record only matched in La Liga this season by Real Valladolid.
Despite making the ideal start, Atletico appeared to lack hunger in attack and began to resemble the timid, conservative style they showed for much of the campaign before the coronavirus disruption.
Their best moment in a flat second half came with a shot from outside the box from Thomas Lemar which was tipped away by Celta's substitute goalkeeper Ivan Villar, who came on late in the game to replace the injured Ruben Blanco.