After Alvaro Gomez brilliantly volleyed the hosts in front, Barcelona clawed their way back to triumph with goals from Ferran Torres, Jules Kounde and Alejandro Balde.
Hoping to bounce back from a painful thrashing by Real Madrid in the Spanish Super Cup final last Sunday in Saudi Arabia, Xavi shuffled his pack.
The coach rested striker Robert Lewandowski and put youngster Marc Guiu in to lead the line, along with Joao Felix.
Guiu, 18, scored the winner for Barça against Athletic Bilbao in La Liga on his debut in October.
The hosts, at their packed 6,000-capacity Reina Sofia stadium, were fired up and dreaming of an upset.
Unionistas beat Villarreal in the last round on penalties, with the match resumed a day after it began but was cut short in extra-time because of floodlight failure.
They spurned a fine chance in the first minute with Mario Losada firing wide after running in on goal.
Guiu had a chance to break the deadlock but his lunging effort was shovelled over by Unionistas goalkeeper Ivan Martin.
When the hosts struck to shock Barcelona it was in spectacular fashion. Juan Serrano crossed to the back post and the unmarked Gomez superbly hammered home on the volley.
Inaki Pena made a smart save at the near post to deny Jorge Rastrojo as the hosts nearly doubled their lead, moments before Barça snatched the equaliser.
Torres broke free on the counter, running through one-on-one from the halfway line, and kept his cool to slide a finish into the bottom corner.
Xavi replaced Andreas Christensen, who was suffering some discomfort, with 16-year-old centre-back Pau Cubarsi at half-time.
The Catalan coach cut a frustrated figure on his touchline and beat the dug-out in anger as his team struggled to impose themselves on the Salamanca minnows.
Kounde put Barcelona ahead with a bolt from the blue in the 69th minute, after being given too much time 25 yards out.
Balde made amends for his poor marking for the Unionistas goal by netting Barcelona's third, a superb solo strike.
The left-back roared into the area and lashed home from a tight angle to wrap up the win.
Pena was forced into a superb double save in the 90th minute as the hosts commendably refused to give up, roared on by their proud fans.
Later Thursday Atletico Madrid host Real Madrid in the final Copa del Rey last 16 clash.
