Pedro and Cristian Tello both struck before the in-form Brazilian made it six goals in three games in all competitions with his late effort at Camp Nou
Leading 4-1 from the first leg, the Catalans put the outcome beyond all doubt when Pedro struck just after the half-hour mark.
Cristian Tello doubled the Blaugrana's advantage with his 68th-minute strike before Neymar maintained his impressive scoring streak with a goal right at the death.
Cartagena faced an unenviable task in trying to turn around a three-goal deficit and their cause was hardly aided by the injury-enforced withdrawal of Eric Lopez after just six minutes of play at Camp Nou, with Fernando coming on in his place.
Barcelona, though, initially struggled to break the visitors down but they finally made the breakthrough in the 31st minute when Pedro swept home a fine cross from the right from Martin Montoya.
The Blaugrana only added a second midway through the second half, in fortuitous circumstances when Tello found the back of the net with a deflected attempted cross.
However, the Catalans sealed victory in fine style, with Neymar heading home his sixth goal in his last three games in all competitions after a terrific run from Andres Iniesta.
Up next for Barcelona is a Liga clash with Getafe on Sunday.