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Valencia, who trail Paris St Germain 2-1 after the first leg of their Champions League last-16 tie, were undone twice in the first 32 minutes at the Romareda with Portugal striker Helder Postiga causing them all kinds of problems.
Apono netted from the penalty spot after Postiga was brought down in the area in the fifth minute.
Postiga then nodded the second after Apono had fired in a free kick.
The visitors found a lifeline when Jonas turned in Aly Cissokho's low cross in the 36th minute and, as Zaragoza tired, the same player volleyed the ball in at the back post after a free kick was headed across the goal in the 69th.
Valencia then hit the crossbar and Zaragoza had reserve goalkeeper Leo Franco sent off for handling outside the area, leaving youth-team keeper Pablo Alcolea to hold the fort for a tense last 13 minutes.
Ernesto Valverde's side have 41 points from 25 games, one behind fourth-placed Malaga who visit Real Betis on Sunday and one ahead of Real Sociedad after their 3-1 victory at Basque rivals Athletic Bilbao on Friday.
Leaders Barcelona, with 65 points, host Sevilla later on Saturday when third-placed Real Madrid, on 49, were visiting basement side Deportivo La Coruna.
Earlier, struggling Real Mallorca slumped to a 3-1 defeat at home to Getafe despite Giovani Dos Santos's outstanding 24th-minute opener.
Diego Castro levelled with a disputed penalty and Adrian Colunga grabbed a double as Getafe scored three times in the last 20 minutes to leave the islanders second from bottom after having won only two points in 2013. (Reporting by Mark Elkington, editing by Tony Jimenez)