The Galicians took the lead through a wonderful goal by Bruno Gama five minutes into the second half, with Nelson Oliveira making sure of the points with his fine lob in stoppage time.
The result delivers coach Fernando Vazquez a first win in five attempts and temporarily lifts his side out of the relegation zone, although they could slip back into it if Real Zaragoza beat Athletic Bilbao later.
Espanyol, meanwhile, waved goodbye to any slim hopes of qualifying for the Europa League and remain 11th in the league table.
Javier Aguirre's Periquitos began the game brightly and could have taken the lead after just three minutes, with Christian Stuani heading a cross from Sergio Garcia off the crossbar before Joan Verdu missed the target from the rebound.
Depor's first proper chance fell to Pizzi after a brilliant exchange with Gama but he tried to lob Espanyol goalkeeper Cristian Alvarez, who got a hand to the shot.
Verdu then tested Dani Aranzubia from long range just before the break but the keeper was equal to the shot.
And the visitors had another great chance to take the lead just after the restart, this time from a counter attack.
Wakaso Mubarak played in Garcia, whose shot was too powerful for Aranzubia to push away, meaning Silvio had to run back and clear the ball off the line.
Three minutes later Gama broke the deadlock at the other end, curling a powerful shot from outside the area into the bottom corner.
Riki should have doubled the lead on 60 minutes but could not beat Alvarez from close range.
Manuel Pablo then nearly turned a Javi Lopez cross into his own net, and Depor got another scare when Verdu went for goal from inside the area, only for Aranzubia to make the save.
Riki was inches away from scoring a brilliant goal after another counter-attack but saw his curled effort go just wide of the post.
The forward was then substituted by Oliveira, who sealed the victory by lobbing Alvarez from just outside the area on the break.
But rather than celebrate, the Portuguese made an angry gesture to the home fans, who have criticised his underwhelming performances this season.