Lionel Messi scored twice for the hosts, while Gerard Pique and Neymar were also on the scoresheet as Luis Enrique's outfit recovered from an early Felipe Caicedo strike to take charge of the affair against Espanyol, who finished with nine men after having both Hernan Perez and Pape Diop sent off in the final 20 minutes.
Arda Turan, who was making his Barcelona debut after being cleared to represent the Catalan side, looked bright in the early moments and set up a half-chance for Neymar inside five minutes, but the Brazilian headed wide, before Sergio Busquets missed the target with a header from the resulting corner.
The home side continued to boss possession as the 10-minute mark approached and again had a half-chance through Neymar after a super pass from Messi, but Javi Lopez and Pau Lopez combined to keep the ball out.
It was Espanyol that opened the scoring in the ninth minute, however, when Caicedo finished from inside the box after Marco Asensio had led a quick counter for the visitors.
Barcelona were level just four minutes later when some slack defending allowed Luis Suarez to rob possession and the striker played in Andres Iniesta, whose pass was finished into the back of the net by Messi.
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Iniesta picked out another pass into Neymar as the home side came close to a second in the 23rd minute, but the Brazilian's low strike flashed across the face of goal, before Caicedo had another half-chance from a header down the other end of the field.
Neymar, Messi and Iniesta continued to impress as the half developed and the former had another shooting chance just past the 30-minute mark, but Javi Lopez, who survived an earlier penalty shout following a late challenge, just did enough to force the Brazilian to fire wide of the post.
Barcelona took the lead for the first time in the match moments before the half-time break, however, when Messi found the back of the net from a free kick after being felled outside the box.
The Spanish champions then scored a third in the 49th minute, Pique tapping home from close range after Messi had put it on a plate for his teammate.
Espanyol goalkeeper Pau Lopez was lucky to stay on the field early in the second half after seemingly standing on Messi's ankle following a coming-together inside the box, and that incident reflected what was another ill-tempered affair between the two Catalan rivals.
Messi had a chance to complete his hat-trick just past the hour, but he could only turn a driven low Neymar cross wide of the post, before Turan stung the palms of Pau Lopez.
Lopez was again in action in the 66th minute as he palmed away a dipping free kick from Neymar, before Aleix Vidal was introduced in the 68th minute to become the home side's second debutant on the night.
It got worse for the visitors in the 72nd minute when Perez was given his marching orders for a late challenge on Alba, which was worthy of a second yellow card in the eyes of the referee. Four minutes later, Espanyol were down to nine when Diop was shown a red for something that he said to Suarez in earshot of the official.
The opportunities continued to fall for Barcelona late on, but Suarez missed a header from close range, before Ivan Rakitic was denied by Espanyol's goalkeeper once again. Neymar was not to be denied, however, as the Brazilian volleyed a Messi pass into the back of the net to make it four for the home team in the 88th minute.
The second leg of the tie will take place next week, but both teams will now switch their attention to La Liga matters. Barcelona will welcome Granada on Saturday, while Espanyol face a trip to tricky Eibar one day later.