After a comfortable first-leg win away, Barca coach Tata Martino took the chance to rest a host of stars but the Catalan giants still had too much class at the Nou Camp.
The home side struggled to find their usual rhythm and survived an early scare before taking a half-time lead when Pedro struck on 31 minutes.
The Spanish winger, who scored twice in the first leg rout, got in front of his marker to neatly tuck home Martin Montoya's cross.
It was better stuff from Barca after the break and they doubled their lead midway through the second half when Cartagena defender Mariano Martinez turned Cristian Tello's cross into his own net.
Brazilian superstar Neymar then continued his recent hot streak, rounding off the scoring two minutes from time when he headed home Andres Iniesta's cross for his sixth goal in his past three appearances.
Barcelona's comfortable win sets up a last 16 clash with Getafe, who brushed aside Girona 4-1 for a 5-2 aggregate victory.
In an all La Liga affair, Osasuna eliminated Malaga on away goals after a 1-1 draw in El Sadar.
Osasuna, who came from three-goals down to draw 3-3 away in the first leg, made the perfect start on Tuesday when Malaga defender Weligton put through an own goal inside the first minute.
The visitors gave themselves hope of a fightback when Eliseu Pereira levelled with a long-range strike on the hour but Osasuna held on to potentially set up a meeting with Real Madrid, who play on Wednesday.
In other results, Villarreal moved through after Jeremy Perbet's second-half goal earned them a 1-0 win at Elche, sending them through 3-2 on aggregate.
Levante easily overturned their 1-0 first-leg loss to Recreativo Huelva, cruising to a 4-0 win at home, while Segunda Division side Alcorcon eliminated Granada on penalties after overturning a 2-0 first leg loss at home.
