A penalty by Apono and a headed goal by Helder Postiga gave the Aragon side a two-goal cushion that was cancelled out by a double from Jonas.
However, things could have been even worse for Manolo Jimenez's side, who played the final 11 minutes of the game with 10 men after goalkeeper Leo Franco was sent off for handling the ball outside his area.
Zaragoza got the ideal start when they were awarded a penalty after just four minutes. Postiga intercepted the ball in the final third and made for the penalty area, where he was tripped by Victor Ruiz.
Apono sent Diego Alves the wrong way from the spot to give his side the lead.
Valencia's first real chance fell to Jonas, but Zaragoza goalkeeper Franco, making his first start of the season, made a smart save to deny the Brazilian forward.
Zaragoza were comfortably the better side in the first half and they doubled their lead just after the hour mark.
Apono delivered a perfect free-kick in from the left-hand side which Postiga sent to the back of the net with an acrobatic diving header.
Jimenez's side should have taken full control after the goal but careless defending let the visitors back in to the game just moments later.
Aly Cissokho got down the left wing and delivered a low cross into the Zaragoza box towards Roberto Soldado, who pulled a dummy, leaving the ball to fall to an unmarked Jonas, who controlled and converted the cross with ease.
Despite conceding, Zaragoza continued to create chances, and Alves had to make two close-range saves, first from Glenn Loovens moments before half-time, and then from Postiga early in the second half.
Nonetheless, Valencia fashioned an equaliser on 69 minutes. Dani Parejo whipped in a free-kick which Ruiz headed towards the far post, and Jonas stuck out a leg to volley the ball in from barely two yards out.
It was then Valencia who took control of the game. Jeremy Mathieu's header was touched on to the bar by Franco, and the goalkeeper was sent off for pushing the ball away from Soldado well outside his area, preventing the Spain international from shooting at an open goal.
Third-choice goalkeeper Pablo Alcolea, on the bench due to the injury to first choice Roberto Jimenez, was brought on with 11 minutes to go and was immediately called into action.
The 23-year-old stopped a powerful strike by Parejo and then miraculously tipped a close range shot by Soldado over the bar to keep the score level.
Zaragoza thought they had won the match in the dying seconds when Postiga bundled the ball into the net after a goalline scramble following a corner, but the goal was ruled out for a foul on Diego Alves.