Barca had to come from a goal down to win the last 32, first leg at the modest club on the south coast, who compete in a division one below Barca's B team, 4-1.
They totally dominated the return at the Nou Camp and breached the massed ranks of the Cartagena defence three times through Pedro's fine 31st-minute effort, Tello's deflected shot with just over 20 minutes left and Neymar's header two minutes from time.
Pedro arrived in the penalty area at the perfect moment to clip a Martin Montoya cross high into the net, before Tello sped on to a Neymar pass and his attempted cross hit defender Mariano and spun past goalkeeper Savu inside the near post.
Spain playmaker Andres Iniesta came off the bench with 15 minutes remaining and it was his perfectly-weighted cross that set up Neymar to score his sixth goal in his last three appearances in all competitions.
Barca, the record winners with 26 Copas, will play top-flight rivals Getafe in the next round after they thumped second-division Girona 4-1 at the Coliseum in the Madrid suburbs to advance 5-2 on aggregate.
Osasuna survived late pressure to see off La Liga rivals Malaga on away goals and will play last season's runners-up Real Madrid or third-tier Olimpic de Xativa in the last 16.
Osasuna came from three goals down to draw 3-3 at Malaga in the first leg and were ahead inside the first minute of the return match at their El Sadar stadium in Pamplona.
Eliseu levelled for Malaga with around half an hour left but the Qatar-owned club, who are struggling in La Liga, were unable to find another goal and the tie ended 4-4 on aggregate.
Villarreal won 1-0 at fellow La Liga side Elche to go through 3-2 on aggregate and Levante thrashed second-division Recreativo 4-0 for a 4-1 success over the two legs.
Real still have work to do at home to Olimpic on Wednesday after the tiny club from near Valencia held them to a 0-0 draw on their artificial pitch in the first leg.
Atletico Madrid, who beat Real 2-1 to win the Cup last season, have a 4-0 advantage when they host third-tier Sant Andreu, also on Wednesday.
Barca look to have an easier path to the final, with Real and Atletico on the other side of the draw and on course to meet in the last four.
The last-16 ties and the quarter-finals are in January, the semi-finals in February and the final is at a yet-to-be-determined venue on April 19.