Valencia were third last season and began with a creditable draw away at Real Madrid, but failed to build on that opening result when they played their first game of the term at the Mestalla.
The draw leaves them 12th in the table with two points.
Granada and Sevilla played out a 1-1 draw in an Andalusian derby at the Estadio Nuevo Los Carmenes.
Sevilla were in trouble from the moment goalkeeper Diego Lopez was sent off just after half an hour in. Andres Palop saved Guillherme Siqueira's penalty, but Mikel Rico gave his side the lead by knocking in the rebound.
The visitors responded with an Alvaro Negredo header just before the interval, and neither side could break the deadlock in the second half.