The Catalans fell behind to Fernando's shock opener for the Segunda Division B side, but responded with two goals from Pedro either side of a Cesc Fabregas strike, before Jean Dongou rounded off matters with the fourth in injury time.
With four away goals to overcome, the Murcian minnows will now be playing for little more than pride during the second leg, which takes place in two weeks' time at Camp Nou.
The likes of Alex Song, Sergi Roberto and Marc Bartra were all handed rare starts, but it was otherwise a strong XI named by Gerardo Martino.
However, matters did not appear to be going according to plan when Fernando beat the defence and Jose Pinto with far too much ease to open the scoring with just over a quarter of an hour played.
However, with less than 10 minutes to go until the break, Pedro levelled for the visitors with an angled strike from Cesc’s through ball.
The former Arsenal skipper then took matters into his own hands just before half-time when he smacked home Adriano's cross to put Barca into a lead that they would never relinquish.
Pedro bagged his second with 15 minutes remaining, racing onto Alexis Sanchez's defence-splitting pass before beating Cartegena keeper Limones to the ball.
The forward nearly completed his hat-trick at the death, but struck the post, only for Dongou to stroke home the rebound to wrap up the victory with his first goal for the club - and the first by an African graduate of La Masia.