The hosts dominated the first half, creating a number of goalscoring opportunities after a somewhat laboured first 20 minutes.
The goalkeeper managed to get back and make a last-ditch save to keep his side on level terms.
After that chance, Blaugrana began to turn the screw and open up the visitors.
Aranzubia made a stunning double save to first deny David Villa and then Tello once more, before Alex Sanchez eventually broke the deadlock in the 38th minute.
That eased the tension around Camp Nou as the teams made their way in for half time.
Barcelona picked up the baton once more after the break but found Deportivo providing a stubborn rearguard.
However, Lionel Messi came off the bench and managed to find a sublime second, which extends his run of scoring in consecutive La Liga games to 17.
He played a neat one-two inside the area before nonchalantly lifting the ball over a helpless Aranzubia.