The hosts needed just two minutes to open the scoring, David Lopez heading them in front from a corner. Diego Colotto got the second goal midway through the first half and Pizzi added a third in the second period, although Juan Rafael Fuentes scored an own goal to give the visitors a late consolation.
The victory means Espanyol can harbour serious ambitions about qualifying for Europe, now that they are just seven points behind sixth-placed Real Sociedad, while Elche are just three points above the bottom three.
Espanyol needed just two minutes to break the deadlock as Sergio Garcia's early shot was deflected out for a corner, and Simao Sabrosa's accurate cross found the unmarked Lopez, who headed straight past Manu Herrera.
Garcia was hungry for a goal of his own and nearly got one a minute after Lopez had scored, but his low shot was saved by Herrera, and Victor Sanchez could only find the side netting from the rebound.
Herrera later blocked a shot from Cristian Stuani but could do nothing about Espanyol's second goal, which arrived when Colotto side-footed another cross into the box from Simao into the net from inside the six yard box.
Elche thought they could have had a penalty at the start of the second half when Coro went down in the area following a challenge from Hector Moreno, while at the other end Pizzi claimed he was tugged around the box but his appeals were shrugged off by the referee.
But the Portuguese's complaints were quickly forgotten when he hit the third goal of the afternoon, turning a defender in the box and then shooting at goal, the ball taking a deflection of Alberto Botia which took it beyond Herrera.
Espanyol stopper Kiko Casilla was forced to make a rare save when Carlos Sanchez tried his luck from the edge of the area, but was unable to stop Elche from scoring in the closing stages.
With 88 minutes on the clock, Fuentes tried to prevent Coro from getting on the end of a corner at the far post but instead diverted the ball into his own net.
But the goal could not take the shine off what was an emphatic win for Espanyol, who toasted their 100th game at their Cornella El Prat stadium with a welcome three points.