كتب : وكالات
Levante's dogged backline made life difficult for Real, who had another clean sheet but struggled to create clear-cut opportunities away from home.
Their best chances fell to Argentinean striker Gonzalo Higuain either side of the interval.
He fired wide with the goal at his mercy early in the first half before wasting another goalscoring opportunity in stoppage time.
He broke clear after a pass from Pedro Leon but could not round the keeper and a Levante defender eventually cleared the danger.
Jose Mourinho's Real, unbeaten so far, have 11 points from five games, two behind leaders Valencia who earlier defeated Sporting Gijon 2-0.
Two first-half goals from Mehmet Topal and Roberto Soldado gave Valencia their fourth victory in five matches.