Joaquin Caparros' side may still be awaiting their first goal of the new Primera Division campaign but they at least banked their first point in front of their home fans - following a 7-0 humiliation at Camp Nou.
Sevilla too earned their first point of the season after they were beaten 3-1 at home to Atletico Madrid last weekend.
Coke headed an Ivan Rakitic corner wide on 20 minutes after a slow start to the match.
Levante began to warm into the contest when Sergio Pinto and Pedro Rios both failed to hit the target from close range but that was the extent of the first-half action.
Sevilla looked more likely after the break with Kevin Gameiro shooting over before Diego Perotti's 18-yarder missed left.
Levante goalkeeper Keylor Navas held on when Rakitic picked out Federico Fazio and the Costa Rican was again called into action to keep out Diego Perotti's right-footed effort from the edge of the area.
Bryan Rabello shot wide when well placed but just as Levante started to look vulnerable they created chances at the other end.
Ruben Garcia drew a save out of Antonio Beto who then punched a 35-yard Sergio Pinto free-kick to safety.
Sergio Garabato forced the last chance of the game wide right of the mark in stoppage time as both sides settled for a point to get their seasons under way.