After Raul Garcia had nodded the visitors ahead after 72 minutes, Postiga finally beat inspired keeper Thibaut Courtois, with the Portugal forward on hand to send a mis-hit Sofiane Feghouli cross into the net from close range in the 93rd.
In new coach Juan Antonio Pizzi's second game in charge, Valencia put on an intense performance at the Mestalla and dominated possession but fell behind when a fluffed clearance by keeper Vicente Guaita fell to midfielder Garcia.
Garcia also had Atletico's best chance of the first half but his rising effort was well saved by Guaita, while Valencia's Juan Bernat drew a fine save from the inspired Courtois with a low drive seven minutes after the break.
Courtois made three more brilliant saves as Valencia pressed for an equaliser, first leaping to tip a curling effort from substitute Sergio Canales around the post in the 76th minute.
The Belgium international, on loan from English Premier League club Chelsea, then denied Pablo Piatti in the 82nd before producing possibly his best stop of the night a minute later from Feghouli's close-range blast.
It looked as if Diego Simeone's Atletico side, who are level with Barcelona at the top of La Liga, would hold on to their lead ahead of the second leg at the Calderon in a week's time but Postiga struck in the final action of the game.
Nevertheless, with an away goal in the bank, Atletico, who beat bitter city rivals Real Madrid 2-1 in last season's final, will be favourites to advance to face Real Betis or Athletic Bilbao, who play on Wednesday, in the quarter-finals.