Atletico scrape draw in Spanish Cup
Earlier, Walter Pandiani's late strike gave Osasuna a valuable away goal as they lost 2-1 at Racing Santander.
In the other two ties, Sevilla took a giant step towards the semi-finals by winning 3-0 at Deportivo La Coruna on Wednesday while Getafe won 2-1 at Real Mallorca.
The second legs are next week.
Celta, who knocked out Villarreal in the last round, were the better side at a packed Calderon, and took a third-minute lead after slicing open the home defence for Roberto Trashorras to score.
Atletico's new Portuguese signing Tiago nodded in from a corner to level in the 11th, but it was a rare sight of goal for them.
Celta controlled possession and forced the hosts into mistakes throughout the game, with only the interventions of Atletico goalkeeper David de Gea denying the visitors a second goal.
At the Sardinero, Uruguayan forward Pandiani grabbed a lifeline for Osasuna on the counter-attack in the 89th minute after his goalkeeper Roberto had been left rooted to the spot by two stunning strikes from Racing.
Gonzalo Colsa scored with a powerful shot from outside the area in the 63rd and Senegalese midfielder Papakouli Diop caught Roberto off his line with a curler from distance in the 84th.