The goalkeeper contrived to punch a second-half corner back across goal to allow Raul Garcia to head home but the Portugal striker nodded in a dramatic late equaliser
Guaita punched a corner back across goal to allow Garcia to head home midway through the second half, as Diego Simeone's side looked set to edge a tight encounter and secure a first-leg advantage.
However, Postia popped up in stoppage time to turn home Sofiane Feghouli's cross and ensure they take a well-earned level scoreline back to the Vicente Calderon for the second leg of the last-16 tie.
Both sides made a number of changes from last weekend's Liga action. Joao Pereira, Andres Guardado, Javi Fuego, Michel and Fede Cartabia and came in for Valencia, while Vicente Guaita got the nod in goal, while Atletico gave a rare start to midfielder Joshua Guilavogui.
Valencia created the first opening of an even first 20 minutes, with Miranda blocking Bernat's effort after Michel's cross, before the full-back then fired a shot wide.
Raul Garcia was then thwarted by Guaita in Atletico's best opening, with the goalkeeper saving superbly after Adrian teed up the midfielder.
Valencia dominated possession but chances remained at a premium for the remainder of the half until the brink of the whistle, when Diego Costa raced clear on the counter thanks to Adrian's pass only for Ricardo Costa to block his effort.
Thibaut Courtois was called into action shortly after the break, diving low to keep out Fede's effort from 20 yards.
Chances remained few and far between, with Valencia unable to find a breakthrough while Atletico waited for the opportunity to counter, but the deadlock was finally broken on 72 minutes.
Guaita made a bizarre mess of an attempted punch from a corner, succeeding only in diverting the ball back across goal to Garcia, who headed into the top corner from point-blank range.
Diego Costa then went into the referee's notebook for diving after going down under a challenge from Ricardo Costa in the penalty are before any contact was made.
Valencia spurned a golden opportunity to draw level when Courtois parried Pablo Piatti's effort out to Canales, but the midfielder failed to get over the ball and turned the rebound over the bar.
Courtois then pulled off another stunning stop to deny Feghouli, who bustled his way past three challenges only to see his fierce effort beaten out.
Atletico remained resolute and looked to have ensured the win until injury time, when Postiga burst onto Feghouli's cut-back to head the ball into the unguarded net for a richly-deserved equaliser which sets up a mouthwatering second leg in the capital.