Luis Suárez and Lionel Messi both scored in the first half to put Barca 2-1 up on aggregate, after Ernesto Valverde's outfit lost the first leg 1-0.
Former Liverpool star Coutinho entered the match looking to help his side ease into the semi-finals of the competition.
Coutinho arrived from Liverpool in a £145 million deal earlier this month but has been recovering from a thigh injury.
The 25-year-old took over the No.14 from Javier Mascherano, who wore the number for seven and a half seasons before his departure from the club was confirmed on Tuesday.
The No.14 at Barcelona has also been worn by Johan Cruyff, Emmanuel Amunike, Gerard Lopez, Santiago Ezquerro and Thierry Henry .
Coutinho's debut came fittingly against a club he once played for, having spent time on loan with Espanyol in 2012.
His first real contribution was drawing a foul in the 70th minute just outside the Espanyol area, with Messi fizzing the ensuing free kick over the bar.
The Brazilian nearly earned his first assist for the club minutes later, slipping a pass into Suarez just in front of goal, only to have the shot brilliantly saved by Pau Lopez, though the Uruguayan really should have finished the chance.
In the end, it was a promising start for the 25-year-old, who had some nice moments in his limited action, and Barcelona did enough to move on to the Copa del Rey semi-finals with a 2-0 win on the night.