In a match that saw precious little goalmouth action, Zaragoza striker Helder Postiga might have claimed only a third La Liga win of the season for his side with a late opening, but he could not take advantage.
At the other end depleted Levante barely threatened and, after their stunning run of seven successive wins earlier this season, they have now picked up only eight points from a possible 30.
For Zaragoza, although they remain seven points adrift of safety and without a win in any competition in over three months, they have at least now picked up two draws in new coach Manolo Jimenez's three games in charge.
Levante had shipped seven goals in their last two games, going down 3-0 at Athletic Bilbao last weekend and then losing 4-1 at city rivals Valencia in the Copa del Rey on Thursday, and coach Juan Ignacio Martinez would have been determined to see a tighter defensive performance from his side on Sunday.
He got that, but there was little from the hosts at the other end as Zaragoza kept only their fourth clean sheet of the season without too much trouble.
Arouna Kone tried his luck for Levante n the 35th minute but saw his 25-yard strike go wide, while Zaragoza's best first-half effort came from Luis Garcia soon after but he too failed to hit the target.
It was a similar story in the second half with Ruben Suarez heading wide for Levante before Zaragoza duo Antonio Apono and Angel Lafita failed to test home goalkeeper Gustavo Munua with their shots.
Defences continued to dominate although Zaragoza created a good chance to take the points four minutes from time but Postiga, having ignored wide-open team-mate Juan Carlos, shot just wide.