The hosts had most of the possession and pressure but it was Espanyol - led for the first time by new coach Javier Aguirre - who had the better opportunities in each half.
Tono made a brilliant save in the Granada goal to deny Christian Stuani the opener moments before half-time, while Raul Baena fired over with the goalkeeper out of position and the goal at his mercy just after the hour mark.
A point for either side means that Espanyol remain rooted to the foot of the table on 10 points with Granada two points and two places better off.
Espanyol became the first side to sack their coach this week as Mauricio Pochettino left following the 2-0 home defeat to Getafe, but Granada boss Juan Antonio Anquela may not be far behind him as the home fans chanted for his departure at the final whistle.
It could have been so different for the hosts had they been more clinical in front of goal, and they had a great chance to take the lead inside the first 60 seconds but Odion Ighalo's shot from 12 yards was blocked away to safety.
Borja Gomez then headed over from a corner while at the other end Tono also had to be alert to smother Juan Forlin's clever effort on 24 minutes.
However it was not until the final minute of the half that the Granada goalkeeper was forced into a serious save as he turned Stuani's header from point-blank range to safety.
Cristian Alvarez was called into action early in the second period by Dani Benitez's free-kick but in the main the major opportunities were Espanyol.
The visitors twice passed up openings to take the lead on 61 minutes as Mubarak Wakaso took a fresh air swipe when clean through but the Ghanaian recovered possession to set up Baena who with fired over an open goal from the edge of the area.
Granada pushed hard for a winner but Alvarez made comfortable saves from Benitez and Jaime Romero before being forced into a good stop in stoppage time from another Benitez free-kick.