The victory for fifth-placed Sevilla at a festive Sanchez Pizjuan trimmed the gap to Bilbao to three points ahead of the Basque club's game at Levante on Monday.
Midfielder Stephane M'Bia powered in a header from a corner to open the scoring for the home side in the 18th minute before Gameiro netted his first of the game moments before halftime when he controlled the ball on his chest and volleyed low into the corner.
Espanyol pulled a goal back two minutes after the break after Sevilla defender Federico Fazio brought down Christian Stuani in the area and Sergio Garcia clipped home the spot kick.
Gameiro's second and Sevilla's third in the 84th minute was the pick of the goals. The Frenchman exchanged passes with Marko Marin and raced into the area before curling a shot past German Parreno in the Espanyol goal.
Sevilla substitute Ivan Rakitic controlled a long pass from the back and fired in at the near post to complete the scoring a minute from time.
"Fourth place will be tough," Sevilla midfielder Victor Machin said on the club website (www.sevillafc.es).
"We are in a good position and why not dream but we have to take each game at a time and keep winning and then we'll see," he added.
Bilbao have 56 points in fourth, which would earn a place in Champions League qualifying, with Sevilla in fifth on 53 in a Europa League berth, the competition they won in 2006 and 2007 when it was called the UEFA Cup.
Sevilla are enjoying another good run in continental competition this term but need to overturn a 1-0 deficit when they host Portuguese side Porto in their Europa League quarter-final, second leg on Thursday.
Valencia, in eighth, lost last week's last-eight, first leg at Basel 3-0 and their chances of a return to the Europa League next season suffered a setback when they were held to a 0-0 draw at struggling Real Valladolid.
Malaga pulled further away from danger after they thumped Andalusian rivals Granada 4-1 and are three points behind Valencia and two adrift of Espanyol and Levante in 11th.
Getafe's chances of avoiding relegation receded when they conceded a goal in the third minute of added time in a 1-0 defeat at fellow strugglers Elche.
Alicante-based Elche have 35 points in 14th, four ahead of Valladolid and Getafe, who are 17th and 18th respectively.
The top three all won comfortably on Saturday and with six games left leaders Atletico Madrid remain on course for a surprise title triumph with 79 points.
Champions Barcelona, who host Atletico on the final day of the season, are a point behind in second, two ahead of third-placed Real Madrid.
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