Cartagena threatened to embarrass their illustrous hosts when Zurdo struck the post midway through the first-half.
However, any hopes the visitors had of a famous victory were quashed when Pedro neatly swept home Martin Montoya's cross for his third goal of the tie.
Without a host of regular starters, Barca struggled to create clear-cut openings all evening but they were gifted a second goal 23 minutes from time when Mariano Martinez turned Cristian Tello's cross into his own net.
And Neymar rounded off the scoring with his sixth goal in his last three games when he headed home Andres Iniesta's cross three minutes from time.
"We played a well and were solid throughout out the game," said Barca boss Gerardo Martino.
"I think we handled the game well, with respect for both the opponent and the people that came to watch."
Martino surprised many by playing Neymar for the full 90 minutes when many of his senior teammates were handed a rest.
However, the Argentine said the 21-year-old will return to Brazil on Thursday for 10 days holiday as he is suspended for Sunday's final game before the winter break away to Getafe.
"He won't play on Sunday so tonight was an opportunity to give him the 90 minutes.
"He's going to Brazil on Thursday. He hasn't had a break since the Confederations Cup so it is good to give him a few extra days."
And Martino also confirmed that he expects the injured Lionel Messi and Victor Valdes to be back training in Barcelona along with Neymar on January 2.
"During the two training sessions on the second we hope to have all the players back together, including those that are continuing to recover like Leo and Victor."
Barca will also face Getafe in the last 16 after the Madrid side thrashed Girona 4-1 to secure their place in the next round by the same score on aggregate.
Osasuna, fresh from their 2-2 draw against Real Madrid on Saturday, will face Los Blancos again in January should Madrid overcome Olimpic Xativa on Wednesday as the Navarrans progressed on away goals after a 1-1 draw at home to Malaga on Tuesday.
Levante hammered Recreativo Huelva 4-0 to also progress 4-1 on aggregate, but the upset of the evening came from second division Alcorcon who overcame a 2-0 first-leg deficit to beat Granada 5-4 on penalties.