Midfielder Gabi Fernandez's shot hit the post, bounced off the back of a prone Alves and landed in the back of the net in the 70th minute at La Romareda stadium.
The hosts had earlier had two shots cleared off the goal line and also hit the posts on two occasions.
Zaragoza defender Paulo Da Silva cleared Pablo Piatti's shot off the line in the 88th to protect Zaragoza's fourth straight home win.
"Tonight we will go to bed and sleep a little more peacefully, nothing more," Zaragoza coach Javier Aguirre said.
Zaragoza playmaker Ander Herrera was sent off for a hard tackle in injury time and will miss next week's match at Real Madrid.
Almeria is 10 points from safety with five rounds to play after its fifth straight loss. Hercules and Osasuna join the Andalusian side in the drop zone.
On Saturday, leader Barcelona beat Osasuna 2-0 to maintain its eight-point lead over Madrid, which thrashed third-place Valencia 6-3. Barcelona visits Madrid on Wednesday for the first leg of the Champions League semifinals.
Athletic Bilbao defeated Real Sociedad 2-1 to move to within six points of fourth-place Villarreal, which lost 3-2 at Sevilla. Atletico Madrid routed Levante 4-1, boosting its bid to qualify for the Europa League bid by opening up a five-point gap over Espanyol.