Sevilla, the 2007 winners, will play Athletic Bilbao for a place in May's final in Valencia, while Barca take on Real Mallorca.
Espanyol had held Barca to a goalless draw in the first leg, but a mistake by Cristian Alvarez gifted the league leaders their opening goal at the Nou Camp on 35 minutes.
The Espanyol goalkeeper dived to block an innocuous-looking shot from Silvinho but could only divert it into the path of Bojan Krkic, who sidefooted into the empty net.
Bojan scored a well-taken second just after the break and defender Gerard Pique made it 3-0 when he nodded home a Daniel Alves cross.
A deflected shot from Ferran Corominas and a spectacular long-range drive from Jose Callejon gave Espanyol hope but they were unable to find the goal that would have taken them through.
They ended the match with 10 men after the referee showed defender Daniel Jarque a second yellow card in added time.
Sebastien Squillaci scored a dramatic last-minute winner as Sevilla came from behind to earn a 2-1 home win over Valencia and edge through on away goals after a 4-4 aggregate.
Valencia looked to be cruising when a header from captain Carlos Marchena put them in front but Frederic Kanoute equalised with a header before the break.
Sevilla missed several chances before Frenchman Squillaci hooked Renato's corner in from close range in the 90th minute.
Coach Manolo Jimenez predicted it would be difficult to get past Athletic in the semi-finals and noted the Basque side had won the Cup 23 times, one fewer than record winners Barca.
"Athletic are just as hard to beat as Valencia, because they are a side with history, they have won a lot of Cups and have been on a long unbeaten run," he told a news conference.
The semi-finals take place over two legs, with the first leg next week and the second in the first week of March.
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