Stephane M'bia netted his first goal in over four months to open the scoring, and Kevin Gameiro doubled the advantage on the stroke of half-time.
That lead was then reduced by Sergio Garcia's penalty at the start of the second half, but Gameiro struck again with five minutes remaining and Ivan Rakitic added to the goals in the final minute.
It was the result Unai Emery's side needed after suffering back-to-back defeats to Celta Vigo and Porto, and sees them move three points clear of Real Sociedad in fifth place, and put pressure on Athletic Bilbao, who play on Monday.
Espanyol made a brighter start and created the first chance of the game, Abraham aiming at goal after receiving a pass from Simao Sabrosa, but the shot was too close to Beto, who saved comfortably.
Sevilla took the lead with 18 minutes gone, M'Bia appearing unmarked to head Trochowski's corner past 21-year-old goalkeeper German Parreno, who stood in for the suspended Kiko Casilla to make his first league appearance of the season.
And the goalkeeper nearly had to pick the ball out of his net again just moments later when Alberto Moreno took aim from the left-hand side of the area, but was relieved to see the ball go wide.
But Sevilla's bombardment on the young stopper's goal resumed a few minutes later, Vitolo crashing a shot wide after combining with Bacca.
Espanyol thought they should have been awarded a penalty when Moreno brought Garcia down on the edge of the area, but the Catalans almost scored from the resulting free-kick, Simao striking the ball through the wall and off the post.
However, Sevilla added another goal before the break, Gameiro chesting down a ball from Vitolo and then driving past German.
Conceding a second goal was a big blow for Javier Aguirre's side but the visitors compensated by scoring right at the start of the second half, Garcia making no mistake from the spot after Christian Stuani had been fouled in the area by Federico Fazio.
Jhon Cordoba had a chance to equalise midway through the second but could only shoot straight at Beto, while Garcia could only shoot into the side netting when he caught a glimpse of goal from the left hand side of the area.
With seven minutes remaining Gameiro made sure of the points, running towards goal from the left wing, playing a one-two with Marko Marin then finishing first time with the inside of his right foot, sending the ball into the bottom corner.
And Rakitic rounded off a fine performance for Emery's side in the 89th minute, controlling a long punt up the field by Nico Pareja with one touch and then firing into the bottom corner.